
Class _TKlll!i 
Book , S5 4 



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COPYRIGHT DEPOSIT. 



The D.Van Nostrand Company 

intend this book to be sold to the Public 
at the advertised price, and supply it to 
the Trade on terms which will not allow^ 
of discount. 



Alternating-Current Machines 



BEING THE SECOND VOLUME OF 

DYNAMO ELECTRIC MACHINERY; 

ITS CONSTRUCTION, DESIGN, 
AND OPERATION 

BY 

SAMUEL SHELDON, A.M., Ph.D. 

PROFESSOR OF PHYSICS AND ELECTRICAL ENGINEERING AT THE POLYTECHNIC 

INSTITUTE OF BROOKLYN, MEMBER OF THE AMERICAN INSTITUTE 

OF ELECTRICAL ENGINEERS, FELLOW OF THE AMERICAN 

ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE, 

AND FELLOW OF THE AMERICAN ELECTRO- • 

THERAPEUTIC ASSOCIATION 

AND 

HOBART MASON, B.S., E.E. 

ASSISTANT IN ELECTRICAL ENGINEERING AT THE POLYTECHNIC INSTITUTE 

OF BROOKLYN, AND ASSOCIATE OF THE AMERICAN INSTITUTE 

OF ELECTRICAL ENGINEERS 



^mi. 




NEW YORK: 

D. VAN NOSTRAND COMPANY 
23 Murray and 27 Warren Sts. 

LONDON: 

CROSBY LOCKWOOD & SON 

7 wStationers' Hall Court, Ludgate Hill 

1902 



.35^ 



THE LIBRARY «F 

e©NGRESS, 
Two Copies Receiveb 

MAY. 8 1902 

ILA88 c^ XXo. NO. 



OL 



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Copyright, 1902, by 
D. VAN NOSTRAND COMPANY 



Typography 



J. Peters & Son. 



Presswork by The F. H. Gilson Company 
Boston, Mass., U. S. A. 



PREFACE. 



This book, like its companion volume on Direct Current 
Machines, is primarily intended as a text-book for use in 
technical educational institutions. It is hoped and be- 
lieved that it will also be of use to those electrical, civil, 
mechanical, and hydraulic engineers who are not perfectly 
familiar with the subject of Alternating Currents, but whose 
work leads them into this field. It is furthermore intended 
for use by those who are earnestly studying the subject 
by themselves, and who have previously acquired some 
proficiency in mathematics. 

There are several methods of treatment of alternating- 
current problems. Any point is susceptible of demonstra- 
tion by each of the methods. The use of all methods in 
connection with every point leads to complexity, and is 
undesirable in a book of this character. In each case that 
method has been chosen which was deemed clearest and 
most concise. No use has been made of the method of 
complex imaginary numbers. 

A thorough understanding of what takes place in an 
alternating-current circuit is not to be easily acquired. It 
is believed, however, that one who has mastered the first 
four chapters of this book will be able to solve any practi- 
cal problem concerning the relations which exist between 
power, electro-motive forces, currents, and their phases in 



IV PREFACE. 

series or multiple alternating-current circuits containing 
resistance, capacity, and inductance. 

The next four chapters are devoted to the construction, 
principle of operation, and behavior of the various types of 
alternating-current machines. Only American machines 
have been considered. 

A large amount of alternating-current apparatus is used 
in connection with plants for the long-distance transmission 
of power. This subject is treated in the ninth chapter. 
The last chapter gives directions for making a variety of 
tests on alternating-current circuits and apparatus. 

No apology is necessary for the introduction of cuts and 
material supplied by the various manufacturing companies. 
The information and ability of their engineers, and the taste 
and skill of their artists, are unsurpassed, and the informa- 
tion supplied by them is not available from other sources. 
For their courteous favors thanks is hereby given. 



CONTENTS. 



CHAPTER PAGE 

I. Properties of Alternating Currents i. 

II. Self-Induction 15 

III. Capacity 29 

IV. Problems on Alternating-Current Circuits. ... 44 
V. Alternators ' . . 57 

VI. The Transformer 92 

VII. Motors 141 

VIII. Converters 169 

IX. Power Transmission '..... 182 

X. Tests 198 



ALTERNATING-CURRENT MACHINES. 



CHAPTER I. 

PROPERTIES OF ALTERNATING CURRENTS. 

1. Definition of an Alternating Current. — An alter- 
nating current of electricity is a current which changes 
its direction of flow at regularly recurring intervals. 
Between these intervals the value of the current may 
vary in any way. In usual practice, the value varies with 
some regularity from zero to a maximum, and decreases 
with the same regularity to zero, then to an equal max- 
imum in the other direction, and finally to zero again. In 
practice, too, the intervals of current flow are very short, 
ranging from -^^ to ^^-^ second. 

2. Frequency. — When, as stated above, a current has 
passed from zero to a maximum in one direction, to zero, 
to a maximum in the other direction, and finally to zero 
again, it is said to have completed one cycle. That is to 
say, it has returned to the condition in which it was first 
considered, both as to value and as to direction, and is 
prepared to repeat the process described, making a second 
cycle. It should be noted that it takes two alteimations . 
to make one cycle. The tilde ( ~ ) is frequently used to 
denote cycles. 



2 ALTERNATING-CURRENT MACHINES. 

The term frequency is applied to the number of cycles 
completed in a unit time, i.e., in one second. Occasionally 
the word alternations is used, in which case, unless other- 
wise specified, the number of alternations per minute is 
meant. Thus the same current is spoken of as having a 
frequency of 25, or as having 3000 alternations. The use 
of the word alternations is condemned by good practice. 
In algebraic notation the letter f usually stands for the 
frequency. 

The frequency of a commercial alternating current 
depends upon the work expected of it. For power a 
low frequency is desirable, particularly for converters. 
The great Niagara power plant uses a frequency of 25. 
Lamps, however, are operated satisfactorily only on fre- 
quencies of 50 or more. Early machines had higher 
frequencies, — 125 and 133 (16,000 alternations) being 
usual, — but these are almost entirely abandoned because 
of their increased losses and their unadaptability to the 
operation of motors and similar apparatus. 

In the Report of the Committee on Standardization of 
the American Institute of Electrical Engineers is the 
following : " In alternating-current circuits, the follow- 
ing approximate frequencies are recommended as de- 
sirable : 

25 or 30 40 60 120 

"These frequencies are already in extensive use, and 
it is deemed advisable to adhere to them as closely as 
possible." 

The frequency of an alternating current is always that 
of the E.M.F. producing it. To find the frequency of the 
pressure or the current produced by any alternating-cur- 



PROPERTIES OF ALTERNATING CURRENTS. 3 

rent generator, if V be the number of revolutions per 
minute, and p be the number of pairs of poles, then 

3. Wave-shape If, in an alternating current, the 

instantaneous values of current be taken as ordinates, and 
time be the abscissae, a 
curve, as in Fig. i, may be 
developed. The length of 
the abscissa for one com- 
plete cycle is— seconds. 

Imagine a small cylinder, 
Fig. 2, carried on one end of a wire, and rotated uniformly 
about the other end in a vertical plane. Imagine a hori- 
zontal beam of parallel rays of hght to be parallel to the 
plane of rotation, and to cast a shadow of the cylinder on 




Fig. I. 




Fig. 2. 



a plane screen perpendicular to the rays. The shadow 
will move up and down, passing from the top of its travel 
to the bottom in a half revolution, and from the bottom 




4 ALTERNATING-CURRENT MACHINES. 

back to the top in another half revolution with a perfect 
harmonic motion. Now imagine the screen to be moved 
horizontally in its own plane with a uniform motion, and 
the positions of the shadow suitably recorded on it, — as 

on sensitized paper or on 
a photographic film, a 
slotted screen protecting 
all but the desired portion 
from exposure. Then the 
trace of the shadow will 
be as in Fig. 3. The 
abscissae of this curve 
may be taken as time, as in the preceding curve, the ab- 
scissa of one complete cycle being the time in seconds of 
one revolution.^ Or, with equal relevancy, the abscissae 
may be expressed in degrees.. Consider the cylinder to be 
in a zero position when the radius to which it is attached 
is horizontal. Then the abscissa of any point is the angle 
which must be turned through in order that the cylinder 
may cast its shadow at that point. In this case the abscissa 
of a complete cycle will be 360°, or 27r. Consideration of 
the manner in which the curve has been formed shows 
that the ordinate of any point is proportional to the sine 
of the abscissa of that point, expressed in degrees. Hence 
this is called a sinusoid or sine curve. 

If the maximum ordinate of this curve be taken as ^„,, 
and time be considered to commence at the beginning of 
any cycle, then the ordinate E' at any time / seconds later , 

will be _ ~^ 

E' = ^,„ sin 2 77/4 

which is equivalent to neglecting ail those intervals of - 
time corresponding to whole cycles, and considering only , 



PROPERTIES OF ALTERNATING CURRENTS. 5 

the time elapsed since the end of the last completed 
cycle. 

As a numerical example : In an alternating-current 
circuit of 45 ^ and a maximum voltage of 100, what 
will be the pressure at 2i seconds after the beginning of 
a cycle } 

E' = 100 sin (2 TT X 45 X 2.125) 

£' . . I 

= sm IQI.2C: TT = sm 1.2 q tt = ~ , 

100 ^ ^ ^ V2 

whence 

E' = — 70.7 volts. 

Since the ordinates of the curve may represent either 
current or pressure, the expression 

I' = I^ sin 277// Y 
is equally true. 

The ideal pressure curve from an alternator is sin- 
usoidal. Commercial alternators, however, do not gene- 
rate true sinusoidal pressures. But the sine curve can 
be treated with relative simplicity, and the curves of 
practice approximate so closely to the sine form, that 
mathematical deductions based on sine curves can with 
propriety be applied to those of practice. Two of these 
actual curves are shown in Fig. 4., 

The shape of the pressure curve is affected by irregular 
distribution of the magnetic flux. Also uneven angular 
velocity of the generator will distort the wave-shape, 
jnaking it, relative to the true curve, lower in the slow 
spots and higher in the fast ones. Again, the magnetic 
reluctance of the armature may vary in different angular 
positions, particularly if the inductors are laid in a few 
large slots. This would cause a periodic variation in the 



6 ALTERNATING-CURRENT MACHINES. 

reluctance of the whole magnetic circuit and a correspond 
ing pulsation of the total magnetic flux. All these influ- 
ences operate at open circuit as well as under load. 



E.M.F. CURVE 

3 PHASE 

40 POLE 

2000 K.W. 

■ 25 

FULLY LOADED 




E.M.F. CURVE 

SINGLE PHASE 

8 POLE 

500 WATTS 

125 'V-' 
NOT LOADED 



Fig. 4. 



There are two other causes which act to distort the 
wave-shape only when under load. For any separately 
excited generator, a change in the resistance or apparent 
resistance of the external circuit will cause a change in the 



PROPERTIES OF ALTERNATING CURRENTS. J 

terminal voltage of the machine. As is explained later, 
the apparent resistance (impedance) of a circuit to alter- 
nating currents depends upon the permeability of the iron 
adjacent to the circuit. Permeability changes with mag- 
netization. Now, because an alternating current is flow- 
ing, the magnetization changes with the changing values 
of current. This, by varying the permeability, sets up a 
pulsation in the impedance and affects the terminal volt- 
age of the machine, periodically distorting the wave of 
pressure from the true sine. 

There are cases of synchronously pulsating resistances. 
The most common is that of the alternating arc. With 
the same arc the apparent resistance of the arc varies in- 
versely as the current. So when operated by alternating 
currents, the resistance of a circuit of arc lamps varies syn- 
chronously, and distorts the pressure wave-shape in a 
manner analogous to the above. 

Summing up, the wave-shape of pressure may be dis- 
torted : At open circuit as well as under load ; by lack of 
uniformity of magnetic distribution, by pulsating of mag- 
netic field, by variation in angular velocity of armature ; 
dindi under load only ; by pulsation of impedance, by pulsa- 
tion of resistance. And the effects of any or all may be 
superimposed. 

4. Effective Values of E.M.F. and of Current One 

ampere of alternating current is a current of such instan- 
taneous values as to have the same heating effect in a con- 
ductor as one ampere of direct current. This somewhat 
arbitrary definition probably arose from the fact that alter- 
nating currents were first commercially employed in light- 
ing circuits, where their utility was measured by the heat 



ALTERXATIXG-CURREXT MACHINES. 



they produced in the filaments ; and further from the fact 
that the only means then at hand of measurmg alternating 
currents were the hot-\\dre instruments and the electro- 
dynamometer, either of which gives the same indication 
for an ampere of direct current or for what is now called 
an ampere of alternating current. 

The heat produced in a conductor carrying a current is 
proportional to the square of the current. In an alternat- 
ing current, whose instantaneous current values vary, the 
instantaneous rate of heating is not proportional to the 
instantaneous value, nor yet to the square of the average 

of the current values, but to the 
square of the instantaneous cur- 
rent value. And so the average 
heatmg effect is proportional to 
the mean of the squares of the 
instantaneous currents. 

The%a'Z'e7'ao-e current of a sinu- 
soidal wave of alternating current, whose maximum value 
is /„,, is equal to the area of one lobe of the curve, Fig. 5, 
divided bv its base line tt. Thus 




/„.. - 



j: 



/„, sin 0^/0 



= —[- cos ^1" =-!„,. 



But the heatino" value of such a current varies, as 



r- = 



Is- 



sin- Ode 



IJ 



sin 2 

4 



/.-. 



The square root of this quantity is called the cfftxtive 

value of the current, / = -^- This has the same heating 

V2 



PROPERTIES OF ALTERNATING CURRENTS. 9 

effect as a direct current /, and the effective values are 
always referred to unless expressly stated otherwise. 
Alternating-current ammeters are designed to read in 
effective amperes. 

Since current is dependent upon the pressure, the 
resistance or apparent resistance of a circuit remain- 
ing constant, it is obvious that if / = ^ then does 

E . 2 ^^ 

also E = ~^- Likewise if average I = - /„, then does also 



V2 



TT 



2 

average E = - E,n. Or these may be demonstrated in a 

TT 

manner analogous to the above. 

The maximum value of pressure is frequently referred 
to in designing alternator armatures, and in calculating 
dielectric strength of insulation. There have arisen vari- 
ous ways of indicating that effective values are meant, 
for instance, the expressions, sq. root of mean sq., V/, 
Vmean square. In England the initials R.M.S. are fre- 
quently used for root mean square. 

^, ^. Effective ^.Jf./^. . n 1 .1 ^ ^ . 

1 he ratio — ^ ,^ _ is called the form-factor, 

Average E.M.F. J J 



since its value depends upon 
the shape of the pressure wave. 
For the curve Fig. 6, the form- 
factor is unity. As a curve be- 
comes ^iore p eaked^ its form- Fig- ^• 

factor increases, due to the superior weight of the squares 
of the longer ordinates. 

In the sinusoid the values found above give 

I 

Form-factor = = i . 1 1 . 



TT 



lO 



ALTERNATING-CURRENT MACHINES. 



Probably no alternators give sine waves, but they ap- 
proach it so nearly that the value i.ii can be used in cal- 
culation without sensible error. 




5. Phase The curves of the pressure and the current 

in a circuit can be plotted together, with theh respective 
ordinates and common abscissae, as in Fig. 7. In some 

cases the zero and the 
maximum values of the 
current curve will occur 
at the same abscissae as 
do those values of the 
pressure curve, as in Fig... 
7. In such a case the 
current is said to be in phase with the pressure. In other 
cases the current will reach a maximum or a zero value at 
a time later than the corresponding values of the pressure,, 
and since the abscissae are indifferently time or degrees, 
the condition is represented in Fig. 8. In such a case,, 
the current is said to be o?(t of phase with, and to lag be- 
hind the pressure. In 

still other cases the / ,'-\---. lagging current 

curves are placed as in 
Fig. 9, and the current 
and pressure are again 
out of phase, but the 
current is said to lead ^^^- ^' 

the pressure. The distance between the zero ordinate of 
one sine curve and the corresponding zero ordinate of 
another, may be measured in degrees, and is called the 
angular displacement or phase difference. This angle of 
lag or of lead is usually represented by ^. When one 




PROPERTIES OF ALTERNATING CURRENTS. II 




curve has its zero ordinate coincident with the maximum 
ordinate of the other, as in Fig. lo, there is a displacement 
of a quarter cycle (</> = 90°), and the curves are said to be 
at right angles. This 

term owes its origin to y -^, \^ leading current 

the fact that the radii 
whose projections will 
trace these curves, as 
in § 3, are at right 
angles to each other. ■^^^- ^' 

If the zero ordinates of the two curves coincide, but the 
positive maximum of one coincides with the negative maxi- 
mum of the other, as in 
Fig. II, then <^ = 180°, 
and the curves are in op- 
posite phase. 

An alternator arranged 
to give a single pressure 
wave to a two-wire circuit is 
said to be a single p/iaser, 
the circuit a single-phase current. 
Some machines are arranged to give pressure to two dis- 
tinct circuits- — each of ^„„^^ 

OPPOSITE PHASE 

which, considered alone, 
is a single-phase circuit 
— but the time of maxi- 
mum pressure in one is 
the time of zero pres- 
sure in the other, so 
that simultaneous pres- 
sure curves from the two circuits take the form of 
Fig. 12. Such is said to be a two-phase or quarter-phase 



>^ — ^ 


RIGHT ANGLES 




<y^=9cr->f , 






/90 


ISOA^ 2>(J'' 360 y 




Fig. 10. 


-^ 


and the 


current in 


the c 




Fig. II. 



12 



ALTERNATING-CURRENT MACHINES. 




system, and the generator is a two-pJiaser. A tJiree-pJiase 
system theoretically has three circuits of two wires each. 
The maximum positive pressure on any circuit is displaced 
from that of either of the other circuits by 120°. As the 

algebraic sum of the cur- 
rents in all these circuits 
(if balanced) is at every in- 
stant equal to zero, the 
three return wires, one on 
each circuit, may be dis- 
pensed wdth, leaving but 
three wires. The three sim- 
ultaneous curves of E.M.F. 
are shown in Fig. 13. The term polyphase applies to any 
system of two or more phases. An ;2-phase system has n 
circuits and n pressures with successive phase differences 

of '^ — ■ degrees. 
n 

6. Power in Alternating-Current Circuits. — \Mth a direct- 
current circuit, the power in the circuit is equal to the 
product of the pressure in volts by the current strength in 
amperes. In an alternating- 
current circuit, the iiistan- 
taiicoiLS power is the product 
of the instantaneous values 
of current strength and 
pressure. If the current 
and pressure are out of 
phase there will be some 
instants when the pressure will have a positive value and 
the current a negative value or vice versa. At such times 
the instantaneous power will be a negative quantity, i.e., 




PROPERTIES OF ALTERNATING CURRENTS. 13 




power is being returned to the generator by the disappear- 
ing magnetic field which had been previously produced by 
the current. This condition is shown in Fig. 14, where 
the power curve has for its ordinates the product of the 
corresponding ordinates of pressure and current. These 
are reduced by multiplying by a constant so as to make 
them of convenient size. 
The circuit, therefore, 
receives power from the 
generator and gives power 
back again in alternating 
pulsations having twice 
the frequency of the gen- 
erator. It is clear that 
the relative magnitudes ^^^' ^^' 

of the negative and positive lobes of the power curve will 
vary for different values of </>, even though the original 
curves maintain the same size and shape. So it follows 
that the power in an alternating-current circuit is not 
merely a function of E and /, as in direct-current circuits, 
but is a function of E, /, and <^, and the relation is deduced 
as follows : — 

Let the accent () denote instantaneous values. If the 
current lag by the angle </>, then from § 3, 

E' = ^,„ sin a, 
where, for convenience, 

a = 2 Try?, 

and r = 1^^^ sin (a — (^). 

Remembering that 

E = — ^, and / = -^ (§4) die instantaneous power, 

V2 V2 

I^' = E' r = 2^/ sin a sin (a - <^). 



14 ALTERNATING-CURRENT MACHINES. 

But sin (a — ^) = sin a cos <^ — cos a sin <^, 

so jP'= 2 ^/(sin^ a cos <^ — sin a cos a sin <f). 

Remembering that <^ is a constant, the average power 
over 1 80°, 

_ 2 £/ cos <^ 



2^7 



cos <^ / '^ . „ , 2^/sin 4> r^" . 

I Sm-^ a^a I sm a COS a^a 

TT Jo TT Jo ■ 

COS ^ fi I . I'^ 2EIsm.d> 

-a sm 2 a 

TT [2 4 



- sm^ a 



F = EI COS cf,. 

Should the current /ead the pressure l)y </)°, then the 
leading equation would be 

F' =■ 2 EI sm a sin (a + ^), 

which gives the same expression, 

E = EI COS ^, 

which is the general expression for power in an alternating- 
current circuit. 

Since, to get the true power in the circuit, the apparent 
power, or volt -amperes, must be multiplied by cos <^, this 
quantity is called \\\^ power factor oi the circuit. If the 
pressure and current are in phase, </) = o°, and the power 
factor is unity. 



SELF-INDUCTION. 15 



CHAPTER II. 

SELF-INDUCTION. 

7. Self -Inductance. — The subject of inductance was 
briefly treated of in § 15, vol. i., of this work ; but, since it 
is an essential part of alternating-current phenomena, it 
will be discussed more fully in thi-s chapter. When lines 
of force are cut by a conductor an E.M.F. is generated in 
that conductor (§ 13, vol. i.). A conductor carrying cur- 
rent is encircled by lines of force. When the current is 
first started in such a conductor, these lines of force must 
be estabhshed. In establishing itself, each line is con- 
sidered as having cut the conductor, or, what is equivalent, 
been cut by the conductor. This notion of lines of force 
is a convenient fiction, designed to render an understand- 
ing of the subject more easy. To account for the E.M.F. 
of self-induction, the encircling lines must be considered 
as cutting the conductor which carries the current that 
establishes them, during their establishment. It may be 
considered that they start from the axis of the conductor 
at the moment of starting the current in the circuit ; that 
they grow in diameter while the current is increasing ; that 
they shrink in diameter when the current is decreasing ; . 
and that all their diameters reduce to zero upon stopping 
the current. At any given current strength the conductor 
is surrounded by many circular lines, the circles having . 
various diameters. Upon decreasing the strength those of 



l6 ALTERNATING-CURRENT MACHINES. 

smaller diameter cut the conductor and disappear into a 
point on the axis of the conductor previous to the cutting 
by those of larger diameter. The number of lines accom- 
panying a large current is greater than the number accom- 
panying a smaller current. 

The E.M.F. of self-induction is always a counter E.M.F. 
By this is meant that its direction is such as to tend to 
prevent the change of current which causes it. When the 
current is started the self -induced pressure tends to oppose 
the flow of the current and prevents its reaching its full 
value immediately, ^^llen the circuit is interrupted the 
E.M.F. of seK-induction tends to keep the current flowing 
in the same direction that it had originall}'. 

8. Unit of Self -Inductance The self-inductance, or 

the coefficient of self-induction of a circuit is generally rep- 
resented by L or /, and is that constant by which the time 
rate of change of the current in a circuit must be multi- 
plied in order to give the E.M.F. induced in that circuit. 
Its absolute value is numerically equal to the number of 
lines of force linked with the circuit, per absolute unit of 
current in the circuit, as is shown below. B}- linkages, or 
number of lines linked with a circuit, is meant the sum 
of the number of hues surrounding each portion of the 
circuit. For instance, a coil of wire consisting of ten 
turns, and threaded completely through b}- twelve lines 
of force, is said to have 1 20 linkages. 

The absolute unit of self -inductance is too small for 
ordinary purposes, and a practical unit, the Jienry, is used. 
This is 10" times as large as the c. g. s. or absolute unit. 

The Paris electrical congress of 1900 adopted as the 
unit of magnetic flux the maxwell, and of flux density the 



SELF-INDUCTION. 17 

gauss. A maxwell is one line of force^ A gauss is one 
line of force per square centimeter. If a core of an electro- 
magnet has a transverse cross-section of 30 sq. cm., and is 
uniformly permeated with 60,000 lines of force, such a 
core may be said to have a flux of 60,000 maxwells and a 
flux density of 2000 gausses. 

In § 1 3, vol. I., it has been shown that the pressure gene- 
rated in a coil of wire when it is cut by lines of force is 

where n is the number of turns in a coil, and where e is ^ 
measured in c. g. s. units, $ in maxwells, and t in seconds. 
In a simple case of self-induction the maxwells set up are 
due solely to the current in the conductor. Now let K be 
a constant, dependent upon the permeability of the mag- 
netic circuit, such that it represents the number of max- 
wells set up per unit current in the electric circuit ; then, 
indicating instantaneous values by prime accents, 

and d^ = Kdi.- 

The E.M.F. of self-induction may then be written 

By the definition of the coefficient of self-induction, 
whose c. G. s. value is represented by /, 

di 
'^ = -'jt' 
From the last two equations, it is seen that / = K71. Kn is. 
evidently the number of linkages per absolute unit current. 
The negative sign indicates that the pressure is counter 
E.M.F, 



l8 ALTERNATING-CURRENT MACHINES. 

In practical units, 

dt 

A circuit having an inductanc e of one henry will have a 
pressure of one volt induced in it by a uniform change of 
current of one a mpere per second. 

9. Practical Values of Inductances. — To give the stu- 
dent an idea of the values of self-inductance met with in 
practice, a number of examples are here cited. 

A pair of copper line wires, say a telephone pole Inie, 
will have from two to four milhenrys (.002 to .004 henrys) 
per mile, according to the distance between them, the 
larger value being for the greater distance. 

The secondary of an induction coil giving a 2^^ spark has 
a resistance of about 6000 ohms and 50 henrys. 

The secondary of a much larger coil has 30,000 ohms 
and about 2000 henrys. 

A telephone call bell with about 75 ohms has 1.5 henrys. 

A coil found very useful in illustrative and quantitive 
experiments in the alternating-current laboratory is of the 
follo\ving dimensions. It is wound on a pasteboard cylinder 
with wooden ends, making a spool 8.5 inches long and 2 
inches internal diameter. This is wound to a depth of 1.5 
inch with No. 16 B. and S. double cotton-covered copper 
wire, there being about 3000 turns in all. A bundle of 
iron wires, 16 inches long, fits loosely in the hole of the 
spool. The resistance of the coil is 10 ohms, and its in- 
ductance without the core is 0.2 henry. With the iron 
core in place and a current of about 0.2 ampere, the induc- 
tance is about 1.75 henrys. This coil is referred to again 
in § II. 



SELF-INDUCTION. IQ 

The inductance of a spool on the field frame of a gene- 
rator is numerically 



o'lf 



where <^ is the total flux from one pole, 71 the number 
of turns per spool, and 7^ the field current of the machine. 
It is evident that the value of L may vary through a wide 
ran2:e with different machines.. 



'fc>' 



$ = 



reluctance 



c 

where c is the mean length in centimeters of the magnetic 
circuit, A its mean cross-sectional area in square centi- 
meters, and /x is permeability. 

Then, if (R stand for the reluctance, 

71 4x;// ^ A /I -n-n^ 

I = -. = 4 Tr7l^ix — 



i c c (Si 

IxJL 
which is independent of i. 



10. Things Which Influence the Magnitude of L. — If all 

the conditions remain constant, save those under considera- 
tion, then the self-inductance of a coil will vary : directly as ^ 
the square of the nu mber of turns ; directly as the linear 
dimension if the coil changes its size without changing its 
shape ; and inversely as the reluctance of the magnetic 
circuit. 

Any of the above relations is apparent from the follow- 
ing equations. The numerical value of the self-induc- 
tance is 

I = 7l~T ' 

As shown in Chapter 2, vol. i., - 

M.M.F. 4 Tr7ii 



20 ALTERNATING-CURRENT MACHINES. 

If, as is generally the case, there is iron in the magnetic 
circuit, it is practically impossible to keep /x constant if any 
of the conditions are altered ; and it is to be particularly 
noted, that with iron in the magnetic circuit, L is by no 
means independent of /. 

II. Growth of Current in an Inductive Circuit. — If a 
constant E.M.F. be applied to the terminals of a circuit 
having b'oth resistance and inductance, the current does 
not instantly assume its full ultimate value, but logarith- 
mically increases to that value. 

At the instant of closing the circuit there is no current 
flowing. Let time be reckoned from this instant. At 
any subsequent instant, t seconds later, the impressed 
E.M.F. may be considered as the sum of two parts, E^ 
and Er. The first, E^, is that part which is opposed to, 
and just neutralizes, the E.M.F. of self-induction, so that 
E^=-E,\ 

but ^^^-^W 

The second part, E,., is that which is necessary to send 
current through the resistance of the circuit, according to 

Ohm's Law, so that 

E, = RI. 

If the impressed E.M.F. 

E = E,^ E^= EI + L— , 

then (E - EI) dt = Idl, 

, ^ L ^^ I Rdl 

and dt = — — ^ dl= — 



EI R E-RI 



SELF-INDUCTION. 



21 



Integrating from the initial conditions t = o, /=o to any 
conditions t = t, I=r, 



L 



\\ogiE-RI')-\ogE\ 



and 



r= 



= log 



j^r 



E 
E 



R 



r' ' =R 



'■) 



where e is the base of the natural system of logarithms. 

This equation shows that the rise of current in such a 
circuit is along a logarithmic curve, as stated, and that when 
/ is of sufficient magnitude to 

render the term e l ^ negli- 
gible the current will follow 
Ohm's Law, a condition that 
agrees with observed facts. 

Fig. 1 5 shows the curve of 
growth of current in the coil 
referred to in § 9. The curve 
is calculated by the above formula for the conditions 
noted. 

The ratio — is called the time constant of the circuit, 
R 

for the greater this ratio is, the longer it takes the current 
to obtain its full ultimate value. 



£- 




^ 




8 




y^ 




7 
CO 6 

*3 


/ 


y^''^ GROWING CURRENT 
/p (direct) E.M..F.= 100 
// R=10 
^ L-.2 




2- 


/ 








1 








) .01 


, 02 . 03 . 04 . 05 . 06 . 07 . 08 

SECONDS 

Fig. 15. 


.09 .1. 



12. Decay of Current in an Inductive Circuit. — If a cur- 
rent be flowing in a circuit containing inductance and re- 
sistance, and the supply of E.M.F. be discontinued, 
without, however, interrupting the continuity of the circuit, 
the current will not cease instantly, but the E.M.F. of 



22 ALTERNATING-CURRENT MACHINES. 

self-induction will keep it flowing for a time, with values 
decreasing according to a logarithmic law. 

An expression for the value of this current at any time, 
t seconds after cutting off the source of impressed 
E.M.F., may be obtained as in the preceding section. Let 
time be reckoned from the instant of interruption of the 

impressed E.M.F. The current at this instant may be 

E 
represented by — , and is due solely to the E.M.F. of self- 
induction. 

Therefore, from Ohm's Law, 

dt 

L dl 
■•■'" = --R^ 

Integrating from the initial conditions t = o, I — ^, to 
the conditions, / = /, 1=1', 



\ 




\ 


DECAYING CURRENT 


\ 


E.M.F.= 


\ 


R=10 


\ 


L=.2 


\. / 


1=10 


\ 


^., 




^ 



:ions t = 


--f 


i>- 


RJE dt 

R 


'=-j 


log-, 




E 


'-f<- 


-i', 



.01 .02 .03 .04 .05 .06 .07 .08 . 09 . 1 i 

SECONDS ana 

Fig. i6. 

which is seen to be the term that had to be subtracted in 
the formula for growth of current. This shows clearly 
that while self-induction prevents the instantaneous attain- 
ment of the normal value of current, there is eventually no 
loss of energy, since what is subtracted from the growing 
current is giver back to the decaying current. 

Fig. 1 6 is the curve of decay of current in the same cir- 



SELF-INDUCTION. 23 

cuit as was considered in Fig. 15. The ordinates of the 
one figure are seen to be complementary to those of 
the other. 

13. Magnetic Energy of a Started Current. — If a cur- 
rent / is flowing under the pressure of E volts, the power 
expenditure is EI watts, and the work performed in the 
interval of time dt is 

dW = Eldt. 



But in a coil of n turns, the E.M.F . induced by a change 



of linkages is 



Substituting, 



F — 

10' dt 



nl 
dW'-= ^d^. 



If the circuit have a constant permeabiHty, 

nd^ = Idi — I o^ Ldl, 
so dW = -Lid I. 

Integrating through the full range, from o to J/Fand from 
o to /, 

dW = - L I Idl, 

Jo 

IV= - \LP. 

Which is an expression for the work done upon the mag- 
netic field in starting the current. When the current is 
stopped the work is done by the field, and the energy is 
returned to the circuit. 

This formula assumes the value of L to be constant dur- 
ing the rise and fall of the current. If there be iron in 
the magnetic circuit the relation nd^ = Idi becomes nd^ 



24 ALTERXATIXG-CURREXT MACHINES. 

= I'di, l being also a variable ; but an average of the val- 
ues of /' throughout the range may be called /, and the 
formula for energ}* stored in the field holds true. 

Smce iron has always a hysteretic loss, some of the 
energy is consumed, and the work given back at the dis- 
appearance of the field is less than that used to establish 
the field by the amount consumed in hysteresis. 

14. Current Produced by a Harmonic E.M.F. in a Cir- 
cuit Having Resistance and Inductance. — Given a circuit 
of resistance R and inductance L upon which is impressed 
a harmonic E.M.F. E of frequency /, to find the current 
/ in that circuit. 

Represent by w the quantity 27r/". 

At any instant of time, /, let the instantaneous value of 
the current be /'. 

To maintain this current requires an E.M.F. whose value 
at this instant is I' R. Represent this by E' ,. 

From § 3, in a harmonic current, 

/' = I„^ sin (o/. 
hence, ^r'= R^,n sin w/. 

Evidently E^ has its maximum va]ue RI,,^ = E^,,^ at uit = <^d'. 
or 270", and its effective value is E, = RI. 

The counter E.M.F . of self-induction at the same instant 
of time, /, is 

— ^' 

But as before, /'= ^ sin cu/. 

so dl' = iiil,,^ cos w/ df. 

and El = — wZv^,, cos wf. 



SELF-INDUCTION. 2$ 

Evidently B^ has a maximum value of — wZ/„, =-Esm at 
o)t = o° or 1 80°, and its effective value ^ 

£, = - oiZl. . 

It is clear that the impressed E.M.F. must be of such 
a value as to neutralize ^5"^ and also supply E^. But these 
two pressures cannot be simply 
added, since the maximum value of. 
one occurs at the zero value of the 
other ; that is, they are at right 
angles to each other, as defined in. 
§ 5. Reference to Fig. 17 will, 
make it clear that combining these at right angles will 
give as a resultant the pressure V^/ + E'^ ; and it is this 
pressure that the impressed E.M.F. E must equal and 
oppose. So 




from which 






\IJ^ + 



This is a formula which must be used in place of Ohm's 
Law when treating inductive circuits carrying harmonic 
currents. It is evident that, if the inductance or the fre- 
quency be negligibly small (direct current has / = o), the 
formula reduces to Ohm's Law ; but for any sensible val- 
ues of (0 and L the current in the circuit will be less than 
that called for by Ohm's Law. 

The expression \IR'^ 4- w^Z^ is called the impeda^ice of 
the circuit, and also the apparent resistance. The term R 
is of course called resistance, while the term wZ, which is 
2 tt/Z, is called the reactartce. Both are measured in ohms. 

The effective value of the counter E.M.F. of self-indue- 



26 ALTERNATING-CURRENT MACHINES. 

tion can be determined as follows, without employing the 
calculus ; that it must be combined at right angles with r 
RI is not directly evident. Disregarding the direction of - 
flow, an alternating current / reaches a maximum value /,„, 
2/ times per second. The maximum number of lines of 
force linked with the circuit on each of these occasions is 
//,„. The interval of time, from when the current is zero 
with no linkages, to when the current is a maximum with 

//„j linkages, is — -^ second. The average rate of cutting - 
4/ 

lines, then, is — ^ , and is equal to the average E.M.F. of • 

4/ 

self-induction during the interval. It has the same value 
during succeeding equal intervals ; i.e., 

4? 
The effective value is (§ 4) therefore, • 

^5 = — 2 -nfli = o)//', 

and in practical units, 

E, = - 2 irfLI. . 

Since the squares of the quantities R, L, and w enter 
into the expression for the impedance, if one, say R, is 
moderately small when compared with L or w, its square 
will be negligibly small when compared with Z" or o>l The 
frequency, because it is a part of o, may be a considerable 
factor in determining the impedance of a circuit. 

Having recourse once again to the harmonic shadow- 
graph described in § 3, the phase relation between im- 
pressed E.M.F. and current may be made plain. It has 
already been shown that E,. and E^ are at right angles to 



SELF-INDUCTION. 



27 



each other. Smce the pressure E,, is the part of the im-^ 
pressed E.M.F. which sends the current, the current must- 
be in phase with it. Therefore there is always a phase 
displacement of 90° between / and E^. This relation is 
also evident from a consideration of the fact that when / 
reaches its maximum value it has, for the instant, no rate 
of change ; hence the flux, which is in phase with the cur- 
rent, is not changing, and consequently the E.M.F. of self- 
induction must be, for the instant, zero. That is, / is maxi- 
mum when Es is zero, which means a displacement of 90°. 
In Fig. 18 the triangle of E.M.F.'?, of Fig. 17 is altered 
to the corresponding parallelogram of E.M.F. 's, and the 
maximum values substituted for the effective. If now the 
parallelogram re- 
volve about the 
center o, the 
traces of the har- 
monic shadows of 
the extremities of 
E,„ ^,,„ and E,^ 
will develop as 

shown. It is evident that the curve E 
curve of current 




Fig. 18. 



and so also the- 
lags behind the curve E' by the angle 
It is clear that the magnitude of </> depends upon the 
relative values of L and R in the circuit, the exact relation 



being derived from the triangle of forces. 



Furthermore 



, E, i^LI 



cos <^ 





2 7r/Z 

R 


Er 




E ' 





that is, the cosine of the angle of lag is equal to the ratio 



28 ALTERNATING-CURRENT MACHINES. 

of the volts actually engaged in sending current to the 
volts impressed in the circuit, and this ratio is again equal 
to the power-factor as stated in § 6. 

15. Choke Coils. — The term choke coil is applied to 
any device designed to utilize counter electromotive force 
of self-induction to cut down the flow of current in an 
alternating-current circuit. Disregarding losses by hyster- 
esis, a choke coil does not absorb any power, except that 
which is due to the current passing through its resistance. 
It can therefore be more economically used than a rheostat 
which would perform the same functions. 

These coils are often used on alternatinsf-current cir- 
cuits in such places as resistances are used on direct- 
current circuits. For instance, in the starting devices 
employed in connection with alternating-current motors, 
the counter E.M.F. of inductance is made to cut down the 
pressure applied at the motor terminals. The starter for 
direct-current motors employs resistance. 

Since a lightni ng discharge is oscillatory in character 
and of enorm ous frequenc y, a coil which would offer a 
negligible impedance to an ordinary alternating current 
will offer a high impedance to a lightning discharge. This 
fact is recognized in the construction of lightning arresters. 
A choke coil of but few turns will offer so great an impe- 
dance to a lightning discharge that the high-tension, high- 
frequency current will find an easier path to the ground 
through an air gap suitably provided than through the 
machinery, and the latter is thus protected. 

Choke coils are also used in connection with alternating- 
current incandescent lamps, to vary the current passing 
through them, and in consequence to vary the brilliancy. 



CAPACITY. 29 



CHAPTER III. 

CAPACITY. 

16. Condensers. — Any two conductors separated by a 
dielectric constitute a condenser. In practice the word is 
generally applied to a collection of thin sheets of metal 
separated by thin sheets of dielectric, every alternate 
metal plate being connected to one terminal of the instru- 
ment, the intervening plates to the other terminal. The 
Leyden jar is also a common form of condenser. 

The office of a condenser is to store electrical energy by 
utilizing the principle of electrostatic induction. If a con- 
tinuous E.M.F. be applied to the terminals of a condenser, 
a current will flow, large at first and gradually diminish- 
ing, till the plates of the condenser have been charged to 
an electrostatic difference of potential that equals and 
opposes the electrodynamic pressure applied. Then there 
is a balance of E.M.F.'?>, and no current will flow if there 
be no leakage. 

A frequent misconception as to the capacity of a con- 
denser is that it is equal to the quantity of electricity it 
will hold. The quantity of electricity a given condenser 
will hold is directly proportional to the tension of the 
charge, and a consideration of this fact leads to the follow- 
ing definition : 

The capacity of a condenser is numerically equal to the 
quantity of electricity with which it must be charged in 



30 ALTERNATING-CURRENT MACHINES. 

order to raise the potential difference between its terminals 
from zero to unity. 

If the quantity and potential be measured in c. g. s. 
units, the capacity, c, will be in c. g. s. units. If practical 
units be employed, the capacity, c, is expressed in farads. 
The farad is the practical unit of capacity. A condenser 
whose potential is raised one volt by a charge of one cou- 
lomb has one farad capacity. The farad is io~^ times the 
absolute unit, and even then is too large to conveniently 
express the magnitudes encountered in practice. The 
term microfarad (-j-o o^"o "o o o" ^^.rad) is in most general 
use. 

In electrostatics, both air and glass are used as dielec- 
trics in condensers ; but the mechanical difficulties of con- 
struction necessitate a low capacity per unit volume, and 
therefore render these substances impracticable in electro- 
dynamic engineering. Mica, although it is expensive and 
difficult of manipulation, is generally used as the dielectric . 
in standard condensers and in those which are intended to 
withstand high voltages. Many commercial condensers 
are made from sheets of tinfoil, alternating with slightly 
larger sheets of paraffined paper. Though not so good as 
mica, paraffin will make a good dielectric if properly 
treated. It is essential that all the moisture be expelled 
from the paraffin when employed in a condenser. If it is 
not, the water particles are alternately attracted and 
repelled by the changes of potential on the contiguous 
plates, till, by a purely mechanical action, a hole is worn 
completely through the dielectric, and the whole condenser 
rendered useless by short-circuit. Ordinary paper almost 
invariably contains small particles of metal, which become 
detached from the calendar rolls used in manufacture. 



CAPACITY. 31 

These occasion short-circuits even when the paper is 
doubled. 

A distinctly different form of condenser is the electro- 
lytic condenser. It consists of two electrodes dipping 
into an electrolyte, as, for instance, two carbon electrodes 
in zinc sulphate. A charge of electricity will deposit 
zinc upon one electrode and set up an E.M.F. of polariza- 
tion. Such condensers should not be subjected to volt- 
ages in excess of their E.M.F. of polarization. Electro- 
lytic condensers have about the same volume as other 
condensers of the same volt-ampere capacity. 

The maximum voltage that may be applied to a con- 
denser is limited by the dielectric strength of the material 
employed. If this limit be exceeded, the dielectric will 
be ruptured, which renders the condenser useless. The 
ohmic resistance of condenser dielectrics is not infinite. 
There is always a leakage from one charged plate to the 
other through the insulation and over its surface. Poor 
insulation may occasion a considerable loss, which appears 
as heat in the apparatus when in use. There is also a 
dielectric hysteresis which is analogous to magnetic hystere- 
sis in iron. A dielectric with a high hysteretic constant 
may consume considerable power when in use, which will 
also appear as heat. 

The capacity of a condenser may be calculated by using 
the following formula : — 

C = .00022 c All , t* 

where 

C = capacity in microfarads, 

A = area of dielectric between two conducting plates 
in square inches, 



32 



ALTERNATING-CURRENT MACHINES. 



71 = number of sheets of dielectric, 
t = thickness of dielectric in mils, 
k = specific inductive capacity of dielectric as obtained 

from the followinsf table. 



Table. 

Glass 3 to 7 

Ebonite 2.2 to 3 

Gutta-percha 2.5 

Paraffin 2 to 2.3 

Shellac 2.75 

]Mica 6.6 

Beeswax 1.8 

Kerosene 2 to 2.5 

17. Connection of Condensers in Parallel and in Series. 

-Condensers may be connected in parallel as in Fig. 19. 

c^ If the capacities of the individual 

condensers be respectively C^, C.„ C^, 

etc., the capacity C of the combina- 

— tion will be 

C=Q+C+ C3+. . . . 

The parallel arrangement of sev- 
eral condensers is equivalent to in- 
creasing the number of plates in one condenser. An 
increase in the number of plates results in an increase in 
the quantity of electricity necessary to raise the potential 
difference between the terminals of the condenser one 
volt ; that is, an increase in the capacity results. 

If the condensers be connected in series, as in Fig. 20,- 
the capacity of the combination will be 

C= ' 



Fig. 19. 



I I 



f 



a 



CAPACITY. 



33 



■ Et >r^- 



— E-2 

— t— - 

Fig. 20. 



For, if a quantity of positive electricity, Q, flow into the 
left side of C^, it will induce and keep bound an equal neg- 
ative quantity on the right side of C^, and will repel an 
equal positive quantity. This last quantity will constitute 
the charge for the r. 

left side of C^. r-C 

The operation is 

repeated in the 

case of each of 

the condensers. It is thus clear that the quantity of 

charge in each condenser is Q. The impressed E.M.F, 

must consist of the sum of the potential differences on the 

separate condensers. Let these differences be respectively 



etc. 



E^, ^2, ^3, etc. 


Then the impressed E.M. 




E = 


: ^1 + ^2 + ^3 + • • • • 


But E^ = 


Q 


-= = l --=1 


and also, 




-1' 


therefore 


Q 
c ' 


c, * c, c. ^ 


or 


c 


I 


k-i*k*- ■■ 



As an example, consider three condensers of respective 
capacities of i, 2, and 5 microfarads. Since the factor to 
reduce to farads will appear on both sides of the equations, 
it may here be omitted. With the three in multiple (Fig. 
19), the capacity of the combination will be 



C- 



2+5=8 mf. 



34 



ALTERNATING-CURRENT MACHINES. 



With the three in series (Fig. 20), 
I 



C 



588 mf. 



Ill 

- + - + - 
I 2 5 

With the two smaller in parallel and in series with the 

larger (Fig. 21), 

I 



C 



I I 

1 + 2 5 



= 1.875 "^f- 



E> 



Fig. 21. 



Cs 

Fig. 22. 



With the two smaller in series and in parallel with the 
larger (Fig. 22), 











c - 


- + 


5 = 5 


666 mf. 










I 


2 






If w 


ith 


any 


condensers 
















Q = c = 


= Q 


= . . 


•=^n 


then, 


W] 


th // 


in 


multiple, 


C = 


nC^, 





and with n in series, 



C 



61. 



It is interesting to note that the formulas for capacities 
in parallel and in series respectively are just the reverse of 
those for resistances in parallel and in series respectively. 

18. Growth of Current in a Condensive Circuit. — The 

opposition to a flow of current which is caused by a con-. 



CAPACITY. 35 

denser is quite different from tliat which is caused by a 
resistance. To be sure, there is some resistance in the 
leads and condenser plates, but this is generally so small 
as to be negligible. The practically infinite resistance of 
the condenser dielectric does not obstruct the current as 
an ordinary resistance is generally considered to do. The 
dielectric is the seat of a polarization E.M.F. which is de- 
veloped by the condenser charge and which grows with it. 
It is a counter E.M.F. ; and when it reaches a value equal 
to that of the impressed voltage, the charging current is 
forced to cease. 

To find the current at any instant of time, /, in a circuit 
(Fig. 23) containing a resistance 7? and a capacity C, the 
constant impressed pressure E must 
be considered as consisting of two 
variable parts, one E^., being active . 
in sending current through the re- 
sistance, and the other part, E^, ■ 
being required to balance the po- 
tential of the condenser. Then at • 

all times 

E = e; + E^. . 

Let time be reckoned from the instant the pressure E is , 

E 
applied ; when, therefore, t = o and I^^ = — . Consider the 

E 

current at any instant of time to be E . Then if it flow- 
for dt seconds it will cause dQ coulombs to traverse the . 
circuit, and 

r=^' or dq = Fdt, 

from which ^, . t-/ , 

dt. 



Q'^j/'' 



36 ALTERNATING-CURRENT MACHINES. 



By definition, 




c- ^ 


therefore, 




,..^J'" 




^' C C 


And by Ohm'e 


5 Law 


5 



so at this instant of time 






c 



-i>' 



which upon differentiating, becomes 



Integrating 



Jo J/o J 

'[log/' 



^C| log /'-log I 



CONDENSER 
CHARGING CURRENT 



E=ioov. T' ^ -^ —-wf- V 
R = 10 ~R ' 

C=2 MF. -''- 

=.000002 F. 




Solving for I\ 



which is the expression 
sought. Like the corre- 
sponding expression for an 
^ inductive circuit, it is losfa- 

Fig. 24. ' Jd 

rithmic. 
Fig. 24 is a curve showing the growth of current in a 
condenser for the conditions indicated. 



) 30 40 50 60 70 80 90 100 
MILLIONTHS OF A SECOND 



CAPACITY. 37 

19. Condensers in Alternating-Current Circuits — Hy- 
draulic Analogy. — Imagine a circuit consisting of a pipe 
through which water is made to flow, first one way, then 
the other, by a piston oscillated pump-like in one section 
of it. The pipe circuit corresponds to an electric circuit, 
the pump to a generator of alternating E.M.F., and the 
flow of water to a flow of alternating current. Further 
imagine one section of the pipe to be enlarged, and in it 
placed a transverse elastic diaphragm. This section cor- 
responds to a condenser. Its capacity with a unit pressure 
of water on one side depends upon the area of the dia- 
phragm, its thinness, and the elastic co-efficients of the ma- 
terial of which it is made. In a condenser the capacity 
depends upon the area of the dielectric under strain, its 
thinness, and the specific inductive capacity of the dielec- 
tric employed. As the w^ater surges to and fro in the 
pipe, some work must be done upon the diaphragm, since 
it is not perfectly elastic. This loss corresponds to the 
loss in a condenser by dielectric hysteresis. The fact that 
the diaphragm is not absolutely impervious to water cor- 
responds to the fact that a dielectric is not an absolute 
electric insulator. As the diaphragm may be burst by too 
great a hydrostatic pressure, so may the dielectric be rup- 
tured by too great an elastic pressure, 

20. Phase Relations. — To understand the relation be- 
tween pressure and current in a condensive circuit, con- 
sider the above analogy. Imagine the diaphragm in its- 
medial position, with equal volumes of water on either side 
of it, and the piston in the middle of its travel. This 
middle point corresponds to zero pressure. When the pis- 
ton is completely depressed, there is a maximum negative 



38 ALTERXATIXG-CURREXT MACHINES. 

pressure, when completely elevated, a maximum positive 
pressure, if pressure and flow upward be considered in the 
positive direction. If the piston oscillate in its path with a 
regidar motion, it is clear that the water will flow upward 
from the extreme lowest to the extreme highest position of 
the piston. That is, there will be flow in the positive direc- 
tion from the maximum negative to the maximum positive 
values of pressure. The direction of flow is seen to 
remain unchanged while the piston passes through its 
middle position or the point of zero pressure. 

Returning to electric phenomena, if a harmonic E.M.F. 
be impressed upon any circuit, a harmonic current will 

flow in it. So in a cir-, 
cuit containins: a con- 
denser and subject to a 
sinusoidal E.M.F., the 
current flow will be sinu- 
soidal. This flow will be- 
^^^' ^^' in the positive direction 

from the negative maximum to the positive maximum of 
pressure, and in a negative direction from the positive 
maximum to the negative maximum, as described above. 
This necessitates that the zero values of current occur at 
the maximum values of pressure ; and since the curves are 
both sinusoids, their relation may be plotted as in Fig. 25. 
It is immediately seen that these curves are at right 
angles, as described in § 5, and that the current leads the 
pressure by 90°. 

Reference again to the hydraulic analogy will show that 
the condenser is completely charged at the instant o£ 
maximum positive pressure, discharged at the instant of 
zero pressure, charged in the opposite direction at the in- 




CAPACITY. 



39 



stant of maximum negative 
pressure, and finally dis- 
charged at the instant of 
the next zero pressure. 
Thus the charge is zero at 
the maximum current flow, 
and at a maximum at zero 
current, that is, when the 
current turns and starts to flow out 
marked in Fig:. 26. 




Fig. 26. 

These points are 



21. Current and Voltage Relations. — If a sinusoidal 
pressure E of frequency/ be impressed upon a condenser, 

the latter is charged in - . - seconds, discharged in the 

I 4 / 

next ■ — seconds, and charged and discharged in the oppo- 

site direction in the equal succeeding intervals. The 
maximum voltage E,n = ^2E (§ 4), hence the quantity at 
full charge is 

Q,,= \J^EC. 

The quantity flowing through the circuit per second is • 

4/G.-4/ V2i^^C. 
This number therefore represents the average current, or 

/,„, = 4 \r^fEC. 
From § 4, the effective current 

2 V2 



whence 
and 



E 



^fCE, 



2 7r/C 



/ 



40 



ALTERNATING-CURRENT MACHINES. 



The last is an expression for the volts necessary to send 
the capacity current through a circuit. The expression 

— is called the capacity reactance of the circuit. It is 

2 ivfC 

analogous to 2 tt/Z, the inductance reactance of an induc- 
tive circuit. 

If the circuit contain both a resistance R and a capa- 
city C, the voltage E impressed upon it must be considered 
as made up of two parts, E,,, which sends current through 
the resistance and is therefore in phase with the current, 
and E^y which balances the counter pressure of the conden- 
ser and is therefore 90° behind the current in phase. 

By Ohm's Law 

E, = RI, 
and from above 

I 



E = 



l-nfC 



I. 



The impressed E must overcome the resultant of these 
two E.M.E.'s; and since they are at right angles 




\Ie,^+ e:^, 



or 



1 = 



Fig, 27. 



V 



7?2 



2 7r/C 



The relation of the E.3I.F.''$, is shown graphically in 
Fig. 27, where the current, which is in phase with the 
pressure E^., is seen to lead the impressed pressure by the 
angle <^. 



22. Resistance, Inductance, and Capacity in an Alter- 
nating-Current Circuit. — The general case of an alter- 



CAPACITY. 



nating-current circuit is one that contains resistance, 
inductance, and capacity. To derive the expression for 
current flow in such a circuit, it is but necessary to com- 
bine the results already found ; and this is most readily 
done graphically, ^n § 14 it was shown that the counter 




>27r/L 




>2T/L 



Figo 280 

E.M.F. due to the inductance reactance of a circuit is 
27r/X, and leads the current by 90°. In §21 it was shown 

that the E.M.F. of capacity reactance of a circuit is 



iizfC 



and lags behind the current by 90°. These two E.M.F. 's 
are, then, in exactly opposite phases, or 180° apart, and 





i_ 

2 7r/c 



Fig. 29. 

the resultant reactance is merely their numerical difference. 
These relations are shown in Fig. 28, where the reactance 
of inductance is greater than that of condensance, and in 
Fig. 29, where the latter exceeds the former, the resistance 
being the same in either case. Clearly the impedance 
resulting from the three factors R, Z, and C is represented 



42 ALTERNATING-CURRENT MACHINES. 

in direction and in magnitude by the hypothenuse as shown, 
and the impressed pressure is proportional to this quantity. 
The general expression for the flow of an alternating 
current through any kind of circuit is therefore 



/ = 



\'^'+ 



irfL 



2 7rfC 



the Cjuantity within the brackets indicating an angle of lag 
of current if positive, and an angle of lead if negative. 

23. Resonance. — If in a circuit containing inductance 
and capacity as well as resistance, the two former are 
proportioned so that 

1 



2 tt/L 



the expression 



reduces to 



1 = 



■ 2 -jt/C' 

E 



V'^+h-iJ 



'--r 

the capacity being of a proper magnitude to balance induc- 
tance. At one instant energy is being stored in the field 
at the same rates it is being given to the circuit by the con- 
denser, and at another instant energy is being released 
from the field at the same rate as it is being stored in the 
condenser. 

When this condition prevails, resonance is said to be 
attained, or the circuit is said to be in tune. 

If the capacity and inductance be in parallel, enormous 



CAPACITY. 43 

currents may flow between the two. This is because the 
two are balanced, and the one is at any time ready to 
receive the energy given up by the other ; and a surging 
once started between them receives periodical increments 
of energy from the Hne. This is analogous to the well- 
known mechanical phenomena that a number of gentle, 
but well-timed, mechanical impulses can set a very heavy^ 
suspended body into violent motion. The frequency of 
these impulses must correspond exactly to the natural 
period of oscillation of the body. 

If the capacity and inductance be in series, the differ- 
ence of potential between the terminals of either may be 
far greater than the E.M.F. impressed upon the circuit. 

In the first case damage is likely to result from the 
overloading of the conductors between the inductance and 
the capacity, even to burning them out, while in the second 
case the pressure may rise to such a point as to puncture 
the insulation of all the apparatus in the circuit, even that 
of the generator itself. " ^ 



44 ALTERNATING-CURRENT MACHINES. 



CHAPTER IV. 

PROBLEMS ON ALTERNATING-CURRENT 
CIRCUITS. 

24. Definitions of Terms. — In considering the flow of 
alternating currents through series circuits and through 
parallel circuits, continual use must be made of various 
expressions, some of which have been defined during the 
development of the previous chapters. For convenience 
the names of all the expressions connected with the general 
equation 



V^^+h^^j^^ 



will be given and defined. 

/ is the current flowing in the circuit. It is expressed 
in amperes, and lags behind or leads the pressure, by an 
angle whose value is 

2 7r/Z 



1 2 TrfC 

<^=-tan-i ^ -^ 

E is the harmonic pressure, of maximum value V2 E, J 
which is applied to the circuit, and has a frequency/. It 
is expressed in volts. 

R is the resistance of the circuit, and is expressed in 
ohms. It is numerically equal to the product of the im- 
pedance by the cosine of <^. 



PROBLEMS. 



45 



L is the inductance of the circuit, and is expressed in 
henrys. 

C is the localized capacity of the circuit, and is expressed 
in farads. ~ ~" 

2 TvfL is the inductive reactance of the circuit, and is 
expressed in ohms. 

is the capacity reactance, or capacitance^ of the circuit, and 
is expressed in ohms. 

[-'"^-^^ 

is the reactance of the circuit, and is expressed in ohms. 
It is numerically equal to the product of the impedance by 
the sine of <^. 



\/^^+h-^^-j^J 



is the impedance or apparent resistance of the circuit, and 
is expressed in ohms. 

T 



the reciprocal of the impedance, is the admittance of the 
circuit. It is expressed in terms of a unit that has never 





Fig. 30. 

been officially named, but which has sometimes been called 
the mho. There are two components of the admittance, 
as shown in Fig. 30. 



46 ALTERNATING-CURRENT MACHINES. 

The conductance of a circuit is that qu antit} ^ by which' 
E must b e multipHed to give t he component of /j2aralleh 
to E. It is measured in the same units as the admittance, 
and is numerically equal to 

cos <^ 



and also to 



impedance 



R^ 



2 77/C^ 



The suscepta nce oi a circuit is that quantity by which 
E m ust be nrultip^lied to give the co mponent of /perpen- 
d icular to E. It is measured in the same units as the 
admittance, and is numerically equal to, 

sin (^ 



and also to 



impedance ' 
I Y 



2 7r/Z 



^fCj 



i?^+ 2 Tt/Z 



2 7r/C, 



It should be noticed that while admittance is the recip- 
rocal of impedance, conductance is not the reciprocal of 
resistance, nor is susceptance the reciprocal of reactance. 
This becomes evident, upon considering numerical values 
in connection with the impedance, right-angled triangle, 
e.g., 3, 4, and 5 for the sides, 

25. E.M.F.'s in Series. — Ahernating E.M.F.'^ that 
may be put in series may differ in magnitude, in frequency, 
in phase relation, and in form or shape of wave. Forms 
other than that of the sinusoid need not be discussed. 



PROBLEMS. 



47 



E.Jf.F.'s of different frequencies in series will give an 
irreo:ular wave-form whose maximum values will recur at 
intervals. The duration of these intervals is the least 
common multiple of the 
durations of the component 
half-cycles. 

If two harmonic E.M.F.'s 
of the same frequency and 
phase be in series, the re- 
sulting E.Iil.F. is merely 
the sum of the separate 
E.M.F.'s. This condition is 
shown in Fig. 31, in which 
the two E.M.F.'s are plotted Fig. 31. 

together, and the resulting E.M.F. plotted by making its 
instantaneous values equal to the sum of the correspond- 
ing instantaneous values of the component E.M.F.'s. The 
maximum of the resultant E.M.F. is evidently 






E,,, = ^1,, + E.,„, 


and since 


E = , E^ = 

V2 V2 


and 


-^2 = ,- ' E = E^ 

V2 



as was stated. 

If two E.M.F.'s of the same frequency, but exactly 
opposite in phase, be placed in series, it may be similarly 
shown that the resultant E.M.F. is the numerical differ- 
ence of the component E.M.F.'s. This case may occur in 
the operation of motors. 

The mxost general case that occurs is that of a number 
of alternating E.M.F.'s of the same frequency, but of 



48 



ALTERNATING-CURRENT MACHINES. 



different magnitudes and phase displacements. The 
changes in magnitude and phase and the "ohise .elation of 
the resulting curve of E.M.F. are shown in Fig. j2, where 
recourse is had once again to the harmonic shadowgraph. 
But two components, E^ and ^.„ are treated, whose phase 
displacement is <^^, The radii vectors E^ra and E,^y„ are 
laid off from o with the proper angle <^^ between them, 
and the shadows traced by their extremities are shown in 
the dotted curves. The instantaneous value of the result- 
ant E.M.F. is the algebraic sum of the corresponding in- 




Fig. 32. 

stantaneous values of the component E.M.F.' ^, and the 
resultant curve of E.M.F. is traced in the figure by the 
solid line. But this solid curve is also the trace of the ex- 
tremity of the line Em, which is the vector sum (the result- 
ant of the force polygon) of the component pressures, E^^ 
and E^rn- This is evident from the fact that any instan- 
taneous value of the resultant pressure curve is the sum of 
the corresponding instantaneous values of the component 
curves, or (§ 3) 

E'= ^1™ sin (Jit + ^2>n sin (w/ + c^^). 
Again from the force polygon 

E^ sin (uit +</))= E^^ sin mt -\- E^^ sin (co/ -j- </>i). 



PROBLEMS. 



49 



Hence at any instant 

E' = E^ sin (cu/ + </)), 

wherefore the extremity of the hne ^„j traces the curve of 
resultant pressure, <^ being its angular displacement from 
E^. If a third component E.M.F. is to be added in series, 
it may be combined with the resultant of the first two in 
an exactly similar manner. 

So it may be stated as a general" proposition, that if any 
number of harmonic E.M.F.'s, of the same frequency, but 
of various magnitudes and 

i<- — £_ — ^ — £_ — ^ — ^^ — ^ 



Fig. 33- 



phase displacements, be 

connected in series, the 

resulting harmonic E.M.F. 

will be given in magnitude 

and phase by the vector sum of the component E.M.F.' ^. 

The analytic expressions for E and (/> may be derived by 

inspection of the diagram, and are 

E = V[^isin<^i+^2sin<jf)2+' • • ]^ + [^i cestui +^2 cos <^2 + - • -J^' 



and 



<^ = 



Ej^ sin 4>i ■+■ E2 sin ^2 + 
E-^ cos <^i -\- E2 cos <^2 + 




Fig. 34. 



As a numerical example, suppose three alternators. Fig. 
33, to be connected in series. Suppose these to give sine 
waves of pressure of values E^ = 70, E.^ = 60, and E^ = 40 



50 



ALTERNATING-CURRENT MACHINES. 



volts respectively. Considering the phase of E^ to be 
the datum phase, let the phase displacements be <^^ = o^, 
<^., = 40°, and <^3 = 75°, respectively. It is required to find 
E and c/). Completing the parallelograms or completing 
the force polygon as shown in Fig. 34, it is found that 
E = 148.7 volts and (/> = 32.1°. 



26. Polygon of Impedances. ^Consider a circuit having 
a number of pieces of apparatus in series, each of which 
may or may not possess resistance, inductance, and 
capacity. There can be but one current in that circuit 
when a pressure is applied, and that current must have 
the same phase throughout the circuit. The pressure at 
the terminals of the various pieces of apparatus, necessary 
to maintain through them this current, may, of course, be 

of different magnitude 
and in the same or 
different phases, being 
dependent upon the 
values of R, L, and 
C. Therefore to de- 
termine the pressure 
necessary to send a 
certain alternatins: cur- 
rent through such a 
series circuit, it is but 
necessary to a dd vec- 
Fig. 35- torially the pressures 

needed to send such a current through the separate parts 
of the circuit. This is readily done graphically. 

Fig. 35 shows the pressures (according to § 22) neces- 
sary to send the current / through several pieces of ap- 




^ PROBLEMS. 51 

paratus, and the combination of these pressures into a 
polygon giving the resultant pressure E necessary to send 
the current / through the several pieces in series. In 
these diagrams, impedance is represented by the letter Z. 
C^ and C^ are l ocalized^ not distributed capacities. 

For practical purposes, the quantity /, which is common 
to each side of the triangle, may be omitted ; and merely 
the impedances may be added vectorially in a " polygon of 
impedances," giving an equivalent impedance, which, when 
multiplied by /, gives E. 

Inspection of the figure shows that the analytical ex- 
pression for the required E is 



The pressure at the terminals of any single part of the 
circuit is 



e^ = isJr^^- 


oiL^ — 


I 12 


E,= I\lRi-^ 


(iiZ.2 — 


I T 


E,= - • ■ • 




It is evident that 







and it is found by experiment that the sum of the potential 
differences, as measured by a voltmeter, in the various 
parts of the circuit, is greater than the impressed pressure. 

27. A Numerical Example Applying to the Arrangement 
Shown in Fig. 35. — Suppose the pieces of apparatus to 
have the following constants ; 



52 



ALTERXATIXG-CURREXT MACHINES. 



i?i = 8^ ohms. Zi = .25 henn-. Q = .000018 farad (18 mf.) 

i?o = 40 ohms. Zo = .3 henry. 

C^ = .000025 farad, 

^i= 100 ohms. 

With a frequency of 60 cycles — whence (0 = 377 — it is 
required to find the pressure necessary to be apphed to the 
circuit to send 10 amperes through it. 







M 


^\ 








7 








/ 








1 


V 




~. / 


T- 






c/ 


rr 






c. / 










"" 








1 






7 


-J 




-^^ 






^ 


^^^ 














^ 


^^ — ■ 




/'' 


- 


-''' 


/ ^ 


^ 


— ' 




L^-^-R^ 


-40 


z. 


= R^ = ioo^ ^^ 




"^ 


■^--~-~-^^' 


= 3; »: 


/ 




\ 


V^^ 




.^ ^/ 
















^^ 


J 


1 
1 








T 


1 


1 








1 
1 
1 


i 








L 






^^■^ 


tsl 




^^- — 


'-''' 


' 


^--''"^ 







R = 225 



P = 11.55" 



Z^ 



229. 



Fig. 36. 

The completion of the successive parallelograms in 
Fig. 36, is equivalent to completing the impedance poly-- 
gon, and the parts are so marked as to require no explana- 
tion. The solution shows that the equivalent impedance, 
^=229.5 ohms, that the equivalent resistance (= actual 
resistance in series), J^ = 22^ ohms, that the equivalent re- 
actance is condensive and equals 46. 2 ohms, and that <^ = 



PROBLEMS. 53 

11.55° of lead. Hence the pressure required to send 10 
amperes through the circuit is 

^ = 10 X 229.5 = 2295 volts. 

To obtain the same results analytically 

^ = 10 V[85 + 4o+ioo]"-^+ [(147-3 — 94-2)- 1 13. 1 + 106. 2]^ 

j^" = 2295 volts. 

The voltages at the terminals of the various pieces of ap- 
paratus are : 

E-^= 10 V85^ + (147.3 ~ 94-2)^ =1001 volts, 
£2. = 10 v4o^ + 113.1^ = 1200 " 

E^ = 10 Vo^ + 106.2^ = 1062 " 

^4=10 Vioo^ + o^ = 1000 " 

^1 + ^2 + ^3 + ^4 = 426^ " 

v^hich is greater than E = 2i()^ volts, showing that the 
mcmerical sum of the pressures is greater than the im- 
pressed pressure ; while the vectorial sum of the separate- 
pressures is equal to the impressed pressure. 

28. Polygon of Admittances. — If a group of several 
impedances, Z^, Z.„ etc., be connected in parallel to a 
common source of harmonic E.M.F. of E volts, their 
equivalent impedance is most easily determined by con- 
sidering their admittances F^, F^, etc . The currents in 
these circuits would be 

I, = EY^. 

The total current, supplied by the source, would be the 
vector sum of these currents, due consideration being given 
to their phase relations. Calling this current /, the equation 
I=EY can be written, where Fis the equivalent admit- 



- B- 



Ey^ 



'■• X-^,^J^.'^-i^^'^'^) '^ ^{J''y^^'<^^'^'^ 



54 



ALTERNATING-CURRENT MACHINES. 



tance of the group. To determine Y, a geometrical addition 
of F^, Fg, etc., must be made, the_ _an^ular relati ons being 
the same as the phase relations of /^ /„ etc., respectively. 
The value of the equivalent admittance may therefore be 
represented by the closing side of a polygon, whose other 
sides are repres ented in m agnitude by the several admit- 
ta nces Y\, F ^, etc., and whose directions are determined- 
by the phase angles of the curren ts /^, 7^, etc., flowing 
through the admittances respectively. The equivalent im- 
pedance then is equal to the reciprocal of F. The sum 
of the instantaneous values of the currents in the branch 
circuits is equal to the corresponding instantaneous values 




Fig. 37. 

in the supply main. As, however, the maximums occur- 
at various times, the sum of the effective currents in the- 
branches is generally greater than the main supply current. 

Fig. 37 is a polygon of admittances, showing the . 
method of obtaining the admittance Fand its phase angle, 
referred to a datum line, which is equivalent to a nimiber- 
of parallel admittances, F^, F^, and F^, with angles <^^, (f).,, ^^\\ 
and </>g, respectively. 

By taking its reciprocal, the equivalent admittance •• 
can be transformed into the equivalent impedance. A 
convenient dimensional scale should be employed. The y 
impedance may be resolved into its equivalent reactance • 



nS 



PROBLEMS. 



55 



C3 



Fig. 38. 



and its equivalent resistance. The equi-" ^' ^'^' 

valent resistance is not the resistance of 

the parallel arrangement as measured 

by direct-current methods. 

As a numerical example, consider the 

same apparatus as was used in the pre- 
ceding example, § 27, to be arranged in parallel, as in Fig. 

38. All the other conditions and values are as stated 

before. It is re- 
quired to find the 
current that will 
flow through the 
mains when ten 
volts are impressed' 
on the circuit. The 
diagram, Fig. 39, is 
self-explanatory. 
The solution shows 
^'^- 39- that the equivalent 

admittance ¥=.0224 and that (/> = 16.1°: From this 

the equivalent impedance 




Z = = 44.6 ohms, )/ 

.0224 



the equivalent reactance 



)Z 



.C 



Z sin cf) = 12.4 ohms, r/ 



and the equivalent resistance R = Z cos 4> = 42.9 ohms. 
The current that will flow under a pressure of 10 
volts is 



I =EY = 10 X .0224 



,224 amperes. 



56 ALTERNATING-CURRENT MACHINES. 

If a circuit have some impedances in series and some 
in parallel, or in any series parallel combination, the 
equivalent impedance can always be found by finding the 
equivalent impedances of the several groups, and then 
combining these equivalent impedances to get the total 
equivalent impedance sought. 



ALTERNATORS. $7 



CHAPTER V. 

ALTERNATORS. 

29. Single-phase Alternators As is the case with 

direct-current machines, alternators have a field and an 
armature. The direct-current machine's commutator is 
replaced, in the single-phaser, by a pair of slip-rings ; and 
the current, instead of being rectified, is lead out as 
alternating current by brushes playing on the rings, as 
described in § 29, vol. i. Revolving field and inductor 
alternators differ from this arrangement, as will be shown 
hereafter. 

It is necessary that all but the very smallest alternators 
should be multipolar to fit them to commercial require- 
ments. For alternators must have in general a frequency 
between 25 and 125 cycles per second; the armature, 
must be large enough to dissipate the heat generated at ■ 
full load without its temperature rising high enough to- 
injure the insulation ; and finally, the peripheral speed of 
the armature cannot safely be made to greatly exceed a 
mile a minute. With these restrictions in mind, and 
knowing that a point on the armature must pass under 
two poles for each cycle, it becomes evident that alterna- 
tors of anything but the smallest capacity must be multi- 
polar. 

In practice it is quite as common to have the field of 
an alternator revolve inside the armature as to have the 



Cl^C/L, /u.. Lcc- - 



f- 



58 ALTERNATING-CURRENT MACHINES. 

armature revolve. In a few instances, notably at Niagara, 
the fields revolve outside the armature. The chief advan- 
tage of the revolving fiejd is that it avoids the collection 
of high-tension currents through brushes^ since the arma- 
ture may be permanently connected up, and only low- 
tension direct current need be fed through the rings to 
the field. Other advantages are increased room for arma- 
ture insulation, and, in polyphasers, the necessity for only 
two instead of three or more slip-rings. 

30. Polyphase Alternators Sing le-phas e_ current s are 

satisfactory for lig'hts, but not, as yet, for pow er. As 
polyphase currents are equally well adapted to both pur- 
poses, and since they are geneially more economical of 
transmission than the single-phase, they are much more 
generally employed. If a motor be operated on a single- 
phase circuit, the supply of power to it is pulsating. 
These pulsations occur with great rapidity, there being in 
the case of unit power factor two for each cycle. A 
single-phase motor must be larger for the same capacity, 
than a polyphase motor. 

Windings for any number of circuits or phases may be • 
placed on a single-armature core, and these may each be 
separately connected to an outside circuit through slip- 
rings, or they may be connected together in the armature * 
according to some scheme whereby one slip-ring will be 
common to two phases. These windings can be placed so - 
that the E.M.F.'s generated therein will have any desired 
phase relations with each other. It is customary to place 
them so that the E.M.F.'s of a two-phase or four-phase . 
system are 90° apart, of a three-phase system are 120° • 
apart, of a six- phase system are 60° apart. . 



ALTERNATORS. 59 

In the following diagrams the curled lines are supposed 
to represent armature windings, which revolve in a bipolar 
field. In some cases they are supposed to be wound on 
cores so as to form pole armatures and in the other cases 
to form ring armatures. The dots at the terminals repre- 
sent points of transition between slip-rings and brushes, 
which are in connection with line wires. It is desirable to 
consider the relations between the E.M.F,!^ generated in 
the armature coils and the pressure between the line-wires, 
as well as between the currents in the armature coils and 
the currents in the line-wires. The assumption is made 
that the different phases are equally loaded, both as to 
current and as to its phase. The system is then said to 
be balanced. It is further assumed that the effective 
E.M.F. in each armature coil is E volts, and the effective 
current / amperes. 

31. Two-phase Systems In the case of two coils and 

four wires, the pressure is E volts between the wires 
attached respectively to each coil.. 
There is no connection between the, bi ' 

two coils and their wires. l.,,./i__. 

In case three wires be employed, as 
shown in Fig. 40, the pressure between 



^ 



m and n or between / and 71 is E ^^^' '*°' 

volts, and ^2E volts between / and m. I amperes flows- 
in / and m and V2/ in ?/., 

32. Four-phase or Quarter-phase Systems When con- 
nected, as in Fig. ^\a, i.e., star comtected, the pressure 
between /and m ox n and / is 2E volts ; between n ox p 
and / or ;;^ it is '^2E volts. The current in each line-wire 
is / amperes. If connected as in Fig ^\b, i.e., ring con- 



6o 



ALTERNATING-CURRENT MACHINES. 



nected^ the pressure is E volts between / and ;/, n and ;;/, 
m and /, or p and /, and ^2E volts between / and in, or 
;2 and/. The current in each line-wire is V2/ amperes. 




Star 




Fig. 41. 



33. Three-phase Systems The pressure and current 

relations in three-phase apparatus are often puzzling to the 
student. Consider three similar coils, x, y, and z, on a 
ring armature, each covering 120°, as in Fig. 4.2a. The 
E.M.F.'^ generated in these coils, when they are rotated 
in a bipola.r field, will have the same maximum values, but 





they will differ in phase from each other by 120°. If two 
of the coils, x and j', be connected as in /;, then the pres- 
sure between the free terminals would be the result of 
adding the two E.M.F.'^ at 120° with each other. If, 
instead of this connection, the one shown in c be made, 
known as the star connection or Y connection, the pres- 
sure between the free terminals Avould be the result of 



ALTERNATORS. 



6l 




Fig. 43. 



siLbtracting the E.M.F. of coil j' from that of .t- at 120°. 

Subtraction is necessary because the connections of coil y 

to the circuit have been reversed. To 

subtract one quantity from another it is 

but necessary to change its sign and 

add. Therefore the pressure between 

the free terminals is that which results 

from adding the E.M.F .'s of x and j/ at 

300° (= 120° + 180°) as shown in Fig. 

43. It is ^^E volts. The star connection is generally 

represented as in Fig. 44, where the pressure between any 

two line-wires is ^E volts, and the current in each 
line-wire is / amperes. 

If the three coils be connected as in 
Fig. 45, the result is termed a delta (A) 
or mesh-connection. The pressure be- 
tween any two of the line-wires is E 
volts. Each line-wire is supplied with 
current from two coils, connection being 
unction between the beginning ( f one coil 
The 

value of the current in each wire 

is V3/ amperes. This results from 

subtracting the current in one coil 

from that in the other at 120°, 

which, as before, is the same as 

adding the currents at 300°. 
The power which is delivered 

by a three-phase machine is not 

altered by changing the method of connection. In one 

case each phase is supplied with / amperes at V3 E volts, 

in the other case with V3 / amperes at E volts. 




Fig. 44. 

made at the 

and the ending of the other 




Fig- 45. 




62 ALTERNATING-CURRENT MACHINES. 

At any instant the current in one wire of a three-phase 
system is equal and opposite to the algebraic sum of the 

currents in the other 
two wires. This is 
clearly shown in Fig. 
46, where the curve 
found by adding at 
each instant the ordi- 
nates of two of the three-phase currents is similar, exactly 
equal, and opposite to the third current. 

34. Electromotive Force Generated In § 13, vol. i., it 

was shown that the pressure generated in an armature is 

V 

^av= 2p^S-—10-\ 
60 

where / = number of pairs of poles,^ 

^ = maxwells of flux per pole, ■ 
V = revolutions per minute, ■ 

and 6^ = number of inductors. 

In an alternating current E = k^E^^^.y where k^ is the 
form-factor, i.e., the ratio of the effective to the average 
E.M.F. Hence in an alternator yielding a sine wave^ 

E.M.F., y 

E = 2.22 p^S^ \0~^. 
60 

V 
Inasmuch 2,?, p — represents the frequency,/, 

E = 2.22 <^^io~^. 

An alternator armature winding may be either concen- 
trated or distributed. If, considering but a single phase, 
there is but one slot per pole, and all the inductors that are 
intended to be under one pole are laid in one slot, then 



ALTERNATORS. 63 

the winding is said to be concentrated, and if the inductors 
are all in series the above formula for E is applicable. If 
now the inductors be not all laid in one slot, but be dis- 
tributed in n more or less closely adjacent slots, the E.M.F. 

generated in the inductors of any one slot will be - of that 

generated in the first case, and the pressures in the differ- 
ent slots will differ slightly in phase from each other, since 
they come under the center of a given pole at different 
times. The phase difference between the E.M.F. gener- 
ated in two conductors which are placed in two successive 
armature slots, depends upon the ratio of the peripheral 
distance between the centers of the slots to the peripheral 
distance between two successive north poles considered as 
360°. This phase difference angle 

width slot 4- width tooth ^ 

</) = —. 360. 

circumterence armature 

no. pairs poles 

If the inductors of four adjacent slots be in series, and 
if the angle of phase difference between the pressures 
generated in the successive ones be <^, then letting E^, E.^ 
E^, and E^ represent the respective pressures, which are 




Fig. 47. 



supposed to be harmonic, the total pressure, E, generated 
in them is equal to the closing side of the polygon as 
shown in Fig. 47. Obviously E < E^ -\- E,^ -\- E.^ -\- E^. 
If the winding had been concentrated, with all the indue- 



64 



ALTERNATING-CURRENT MACHINES. 



tors in one slot, the total pressure generated would have 
been equal to the algebraic sum. 

The ratio of the vector sum to the algebraic sum of 
the pressures generated per pole and per phase is called the 
distribittiofi constant. Not only may the number of slots 
under the pole vary, but they may be spaced so as to 
occupy the whole surface of the armature between succes- 
sive pole centers (the peripheral distance between two 
poles is termed the pole distance), or they may be crowded 

together so as to 



1.00 



.99 


^ 


^ 
















Q8 








^ 












.97 








> 


^ 


V 








.96 










^ 


^ 








95 












^ 


^ 






.94 














% 


\ 




.93 














\ 


^ 


\ 


.92 
















\ 


V^ 


.91 


















\^^ 


.90 




F 


raction of Pole Distance 
Occupied by piots.| 




\ 



1 Slot. 



2 Slots. 



3 Slots. 

4 Slots. 
Many 
Slots. 



occupy only one- 
half, one-fourth, or 
any other fraction 
of this space. Both 
the number of slots 
and the fractional 
part of the pole dis- 
tance which they 
occupy affect the 
value of the distri- 
bution constant. A 
set of curves. Fig. 
48, has been drawn, 
showing the values of this constant for various conditions. 
Curves are drawn for one slot (concentrated winding), 2, 
3, 4 slots in a group, and many slots (i.e., sm.ooth core 
with wires in close contact on the surface). The ordinates 
are the distribution constants, and the abscissae the frac- 
tional part of the pole distance occupied by the slots. 

The distribution constant, k,^, must be introduced into 
the formula for the E.M.F. giving 

V 
60 



Fig. 48. 



ALTERNATORS. 



65 



or, for sine waves, 

E = 2.22 k^^Sf\o-^. 

35. Armature Windings. — Some simple diagrams of 
the windings of multipolar alternators are given in Fig. 49 
ct seq. The first is a single-phase concentrated winding, 
with the winding which is necessary to make it two-phase 
in dotted Hnes. If the two windings be electrically con- 
nected where they cross at point P the machine becomes 




/ 4- POLE. \ 

/ SINGLE PHASE. \ 

CONCENTRATED. 
ADDITION OF DOTTED WINDING MAKES IT 
TWO PHASE 




Fig. 49. 

a star-connected four-phaser. Fig. 50 is a three-phase, 
A connected, concentrated winding. Fig. 5 i is the same 
but Y connected. The common junction of the windings 
would have to be provided with a slip-ring if it were 
desired to operate a three-phase, four-wire system with 
the fourth wire connected to the machine. Fig. 52 is a 
three-phase, A connected winding distributed over two 
slots. In all these diagrams the radial lines represent the 
inductors ; other lines the connecting wires. The induc- 
tors of different phases are drawn differently for clearness. 



66 



ALTERNATING-CURRENT MACHINES. 



Where but one inductor is shown, in practice there would 
be a number wound into a coil and placed in the one slot. 
For simplicity all the inductors of one phase are shown in 
series. In concentrated windings, all inductors of one 




4- POLE. 

3- PHASE. 

Y. 

CONCENTRATED. 

Fig. 51. 



phase carrying current in the same direction could be 
connected in multiple if desired ; but with distributed wind- 
ings, the coils cannot all be placed in multiple, because the 
small phase differences between them would set up local 
currents and give rise to undue heating. 

To determine the interior connections for a three-phase 
A winding, place the inductors of a coil of one phase under 
the centers of the poles, then a maximum pressure in a 
given direction is generated therein. Since the algebraic 
sum of the pressures around the A must be zero, the other 
two phases must be connected so that their pressures 
oppose the first. To determine the y connection, place 
the inductors of one phase under the centers of the poles. 
The E.M.F. of this phase will now be at a maximum, say, 
away from the common center. The other two phases 




ALTERNATORS. 6/ 

must be so connected as to have E.M.F.'s toward the com- 
mon center at this instant. 

36. Armature Reaction The armature reaction of an 

alternator consists of two parts, distortion and magnetiza- 
tion or demagnetization. These depend upon the arma- 
ture ampere-turns and upon the lag or lead of the armature 
current. The maximum pres- 
sure is generated in a coil when 
its opposite inductors are re- 
spectively under the centers of 
north and south poles. This con- 
dition is represented in Fig. 53. 
If the armature current be in phase with the pressure, /„, 
in the coils coincides with Ey^, and poles on the armature 
are formed as shown. It is seen that the M.AI.F.'s, both 
of the field and of the armature conspire to concentrate 
the flux in the trailing pole tips. So with / in phase with 
E the armature M.M.F. chiefly effects a distortion of the 
lines, entailing a greater flux density, hence a lower per- 
meability, and also a greater length of air-gap path. This 
slightly decreases the flux, and affects the regulation of 
the alternator. 

If, now, the current be lagging, the armature will have 
reached a position in advance, at the instant of maximum 
current. Therefore, like poles of the field and of the 
armature will be more directly opposite to each other. 
The distorting influence will be present in a degree ; and 
there will be considerable demagnetization of the field, due 
to the opposing M.M.F.'?, of the armature and field ampere- 
turns. If the current be leading, then, at the instant of 
maximum current, a south armature pole will be more 



68 ALTERXATING-CURREXT MACHINES. 

nearly opposite to a north field pole, and their M.Jf.F.'s 
will be cumulative. The field will be strengthened if the 
magnetizing reaction exceeds in effect the skewing reac- 
tion. Alternators have a much better regulation on non- 
inductive loads than on inductive loads. 

37. Armature Inductance The impedance of an alter- 
nator armature is made up of its ohmic resistance, J^, 
combined at right angles with its reactance, 2 tt/L. In 
practice the inductance, L, is likely to be so great that i? 
becomes negligible, and the impedance equals the reac- 
tance. The armature reactance may or may not be an 
appreciable part of the impedance offered by the completed 
circuit. If it is appreciable, then the current in the circuit 
will lag even with a non-inductive load. In any case there 
win be loss of voltage due to armature impedance which 
(when R is negligible) is equal to 2 tt/Z/. This is at right 
angles to the current, and must be properly combined with 
/ times the equivalent impedance of the external cir- 
cuit to determine the pressure actually generated in the 
machine. In special cases the armature reactance is the 
predominant feature of the circuit ; for instance, alternators 
for series arc lighting are made with so great a reactance 
that the impedance of the external circuit within the 
limits of operation is negligible in comparison. The altera- 
tion in the value of this impedance does not, then, appre- 
ciably alter the total impedance of the circuit, and the 
alternator therefore operates as a constant-current gen- 
erator. Many commercial alternators have sufficient arma- 
ture reactance to prevent their injuring themselves on 
dead short circuit for a limited time. It is necessary 
that armatures should have some considerable inductance 



ALTERNATORS. 69 

when alternators are to be operated satisfactorily in 
parallel. 

38. Synchronous Reactance When an alternator is op- 
erating on a load, the pressure, which would be generated 
on open circuit at the same speed and excitation, is made 
up of the following parts, and might be found by adding 
them together in their proper phase relations : 

(a) terminal voltage, E, 

(d) ohmic drop in armature, IR, in phase with the cur- 
rent, 

(c) armature inductance drop, 90° with the current, 

(d) deficit of actually generated volts due to increase of 
magnetic reluctance accompanying distortion, 

(e) deficit or increment of actually generated volts due 
to the demagnetization of a lagging current or the mag- 
netization of a leading current. 

All the parts, except the first mentioned, can be grouped 
together, and be dealt with collectively by the use of a 
quantity called the synchronoiLS impedance. It is that im- 
pedance, which, if connected in series with the outside cir- 
cuit and an impressed voltage of the same value as the 
open-circuit voltage at the given speed and excitation, would 
permit a current of the same value to flow as does flow. 
This quantity for any load can be determined experimen- 
tally with ease. The synchronous impedance has two fac- 
tors, namely, the armature resistance and a quantity termed 
the synchronous reactance. The two, when combined at 
right angles, give the synchronous impedance. 

Since the synchronous impedance takes account of all 
the diverse causes of voltage drop above enumerated, it is 
clear that it has not a physical existence, but is merely a 



70 



lXg-currext machines. 



fictkm. It li ; _^ . : -e in determining the perfcnnaiice 
of a machine. Its Taiue is the same isx all exdtatioiis o€ 
the field, but is somewliat different for \arioas kndsw 
These two facts afford a Teij ccmYenient means of deter- 
mining its ¥alue: Rnn the altemator at its premier ^peed. 
Shcxt-drcoit the aimatnre through an ammeter. Ex.ire 
the fidd until the ammeter indicates the desired itujio. 
Then o/^tok the load drcoit and read the tanninal Toitagei 
The quotient ctf the vcdts by the amperes is the spM^ron- 
ons impedance. It maj happen that the resistance dt the 
armature is neghgifaly small, in whidi case the svndbron- 
oos reactance equals the syncfarcmous impedance. 



39. Satoratioa OocfficigBt, — A noJoad saturation curve 
of an altemator may be obtained by measuring the termi- 
nal vxihage corre^Mxiding to various stioigths of field cur- 
rent, when the machine is running at its proper ^peed and 

without load. Laying 5 
EJifJ\% E as onfcj^ic^ 
and exciting currents^ Ij^ 
as abscissae, a curve is 
found as in Fig. 54. 
dl. 

The latio -f- is called 
dE 

rbe no4oad satnratum co- 

"- - eficiemt erf the machine^ 

Another : '^^ the load-saturation curve can be 

obtained by ^ uUe non4a<luctive resbtance for 

maintaining the constant full load. The terminal volts 

owre^ionding to vair" .15 -.-\' 7 "ations are read cm a 




ALTERNATORS. 71 

voltmeter. This cune will approximately parallel the no- 
load saturation cur\-e. It wiQ have a zero voltage value 
for that excitation which causes sufficient voltage to send 
the full-load current through the synchronous impedance 
of the armature. A full-load saturation coefficient cur^e 
might be drawn from the full-load saturation curve. It 
wiU nearly coincide with the other coefficient cune. 

These saturation curves have forms similar to masaieti- 
zation curves for iron. The knee, however, is less abrupt 
than is £:eneral in an iron curve, because of the unvar\-inor 
permeability of air, and because the different magnetic 
parts of the generator do not reach saturation at the same 
time. If the alternator is normally excited to above the 
knee of the saturation curve, it will require a considerable 
increase of field current to maintain the terminal voltage 
when the load is thrown on, while if normally excited 
below the knee, a slight increase of excitation will suffice. 
The regulation is, however, better when the magnetization 
is abo\'e the knee ; that is, with unaltered field strength, 
the voltage rise upon throwing off the load is less than if 
the excitation were below the knee. 

40. Leakage Coefficient As in direct-current machines, 

the leakage coefficient of an alternator may be defined as 
the number of maxwells set up by the field divided by the 
number of maxwells passing through the armature. It is 
always greater than unity. Its value depends upon the 
design of the machine, upon the pemieability of the various 
parts makmg up the magnetic circuit, upon the load on 
the machine, and upon the degree of saturation in the 
fields. In modern commercial machines of size its values 
lie between i.i and 1.5. 



72 ALTERNATING-CURRENT :.:a::-::nes. 

41. Efficiency The following is abstracted from the 

Report of the Committee on Standardization of the Ameri- 
can Institute of Electrical Engineers. Only those por- 
tions are given which bear upon the efficiency of alternators. 
They wfll, however, apply equall)" well to s}Tichronous 
motors. 

The " efficiency" of an apparatus is the ratio of its net 
power output to its gross power input. 

Electric power should be measured at the terminals of 
the apparatus. 

In determining the efficiency of alternating-current 
appa::.: :5. the electric power should be measured when 
the cuTTr : :- : hase with the E.M.F. unless otherwise 
specified. c:::c;^: vhen a definite phase difference is in- 
here": in the apparatus, as in induction motors, etc. 

Wliere a machine has auxiliar)^ apparatus, such as an 
exciter, the power lost in the auxOiar}- apparatus should 
not be charged to the machine, but to the plant consisting 
c: tie machine and auxiliar}' apparatus taken together. 
The plant efficiency in such cases should be distinguished 
from the machine efficienc}'. 

The efficiency may be determined by measuring all the 
losses indiriduaUy, and adding their sum to the output to 
derive the input, or subtracting their sum from the input 
to deri\e the output. AH losses should be measured at, 
or reduced to, the temperature assume:. i:i continuous 
operation, or in operation under conditions specified. 

In s\Tichronous machines the output or input should be 
measured with the current in phase with the terminal 
E.M.F. except when othervi-ise expressly specified. 

Owing to the uncertainty" necessarily involved iri the 
approximation of load losses, it is preferable, whenever 



ALTERNATORS. 73 

possible, to determine the efficiency of synchronous ma- 
chines by input and output tests. 

The losses in synchronous machines are : 

a. Bearing friction and windage. 

b. Molecular magnetic friction and eddy currents in 
iron, copper, and other metallic parts. These losses should 
be determined at open circuit of the machine at the rated 
speed and at the rated voltage, + /i? in a synchronous 
generator, — //? in a synchronous motor, where / = cur- 
rent in armature, R = armature resistance. It is undesir- 
able to compute these losses from observations made at 
other speeds or voltages. 

These losses may be determined by either driving the 
machine by a motor, or by running it as a synchronous 
motor, and adjusting its fields so as to get minimum cur- 
rent input, and measuring the input by wattmeter. The 
former is the preferable method, and in polyphase ma- 
chines the latter method is liable to give erroneous results 
in consequence of unequal distribution of currents in the 
different circuits caused by inequalities of the impedance 
of connecting leads, etc. 

c. Armature-resistance loss, which may be expressed 
by/ I^R ; where R = resistance of one armature circuit 
or branch, / = the current in such armature circuit or 
branch, and / = the number of armature circuits or 
branches. 

d. Load losses. While these losses cannot well be 
determined individually, they may be considerable, and, 
therefore, their joint influence should be determined by 
observation. This can be done by operating the machine 
on short circuit and at full-load current, that is, by deter- 
mining what may be called the ''short-circuit core loss." 



74 ALTERNATING-CURRENT MACHINES. 

With the low field intensity and great lag of current 
existing in this case, the load losses are usually greatly 
exaggerated. 

One-third of the short-circuit core loss may, as an 
approximation, and in the absence of more accurate infor- 
mation, be assumed as the load loss. 

e. Collector-ring friction and contact resistance. These 
are generally negligible, except in machines of extremely 
low voltasre. 

/. Field excitation. In separately excited machines, 
the I'^R of the field coils proper should be used. In self- 
exciting machines, however, the loss in the field rheostat 
should be included. 

42. Regulation for Constant Potential. — Alternators 
feeding light circuits must be closely regulated to give 
satisfactory service. The pressure can be maintained 
constant in a circuit by a series boosting transformer, but 
it is generally considered better to regulate the dynamo by 
suitable alteration of the field strength. 

The simplest method of regulating the potential is to 
have a hand-operated rheostat in the field circuit of the 
alternator, when the latter is to be excited from a com- 
mon source of direct current, or in the field circuit of the 
exciter, if the alternator is provided with one. The 
latter method is generally employed in large machines, 
since the exciter field current is small, while the alternator 
field current may be of considerable magnitude, and would 
give a large I'^R loss if passed through a rheostat. 

A second method of regulation employs a composite 
winding, analogous to the compound windings of direct- 
current generators. This consists of a set of coils ; one 



ALTERNATORS. 



75 



on each pole. These are connected in series, and carry 
a portion of the armature current which has been rectified. 
The rectifier consists of a commutator, having as many 
segments as there are field poles. The alternate segments 
are connected together, forming two groups. The groups 
are connected respectively with the two ends of a resis- 
tance forming part of the armature circuit. Brushes, 



ys a-eo-900 roi 

^S S-90-900 To, 
»qs O-120-900 Fo' 



T^S l2-i00-S00ForTT> i^ 
T^S 20-l©0-750 Forrr. jq 




Cornmutator-Collectorj 



manner or placing spools 
The observer Ts supposed to be loohtn^ at faces 
or pole pieces n^arhed A and B. The series field 

■the armature -that Is. 
arro^^S corresponc* t-O 
those on spool -flanges, the spools being so placad 
, that the arrows point In opposite dJrect'ons on 

.Collector side each.succeeci'ng spool»_- 



ye 



ling sHould be nea 



Fig. 55. 



bearing upon the commutator, connect with the terminals 
of the composite winding. The magnetomotive force of 
the composite winding is used for regulation only, the 
main excitation being supplied by an ordinary separately 
excited field winding. The rectified current in the com- 
posite coils is a pulsating unidirectional current, that 
increases the magnetizing force in the fields as the cur- 
rent in the armature increases. The rate of increase is 



•^ AI-TZRXATIXt(i-^URREXT 3i£A~HIXZ5, 



ddtemuned bv the issisiaiice oft a simnt placed aooss tibe 

bn^ies r ~ —oreasH^ tre rftpiftsiBce of tii^ shunt, the 

aonKumait — i" ™mglm^ -:-- le nacreased. Widi sodt 

iJi- irri. . jltematcr can be cpc^r-compoandBd to 

T~::oilage of potcntBl drop ni the 

: _ 7 7 --r^l^iod here CNilfined E5 osedbj 

: -T T-T t::^ Z-t;".\. . znjraisv in tibder san^e-filiase 

-y 6eid aitdnatL: r^ Tit : jimections are s':: :^-z 

^ - 55- 

-' — r"'->(id ctf ¥Pg THlbiriifMi is oDopia^inQd bf the Weslt- 
. :^nF <» their lew^cin^ annatme alter- 

naccKs, ene at wMdi, a j^ ^w_, 6o^, sii^e-fihase madnne^ 
is shoPHii in Fig. 56. A composite windii^ is eni^ofied, 
and the compensaffex ~~r?? ^r^ (r\«Tird by ciuieut from 
a 'ieries tiansiomTr : i± ^^okes itf the arantme 

spider. The pr ^ ir.£fflM<aoner consissts of but 

a. few turns;, ani£ t ^rmaSufe current b conducted 

throi^^ it befe-T _ zhet oafifector lii^^ The sec- 

ondaiy of Idns tian^onner ts> suitably (connected to a 
ample conmnttator on the extreme end of the shaft. 
Upon this lest the bm^bes vlaich aie attacfafad to the ends 
of the compensating ooiL This comnmtalor is sufafecfed 
to €»ilf moderate currents and lew Tuitages. The cunent 
in the secondary of the transformer, ai»d hence that in 
the oooqiensatii^ cxmI, is pnoportiomal to tlae mcm% armature 
OKient. The machine is wound f«or tlitc ^;'.~^-:ii3im dear- 
able ocer-cannpoiimdnig, 2taid any less .an be 
secured 1^ sli^tlT ^nftir^ the ^— __—: : -~fSi. 
For there are only as many se^;inents as pofies : : i^e 
brushes ^nn the msulation p^t when the wa\ie ot csurcait 
in the tiansioimer secondbry is pa^ sm n^ throii^ zero^ 
thepulsatii^ direct c ur r ent in the conqioundii^ coil 



ALTERNATORS. 



77 



is equal to the effective value of the alternating" current ; 
but if the brushes are at some other position, the current 
will flow in the field coil in one direction for a portion of 
the half cycle, and in the other direction for the remaining 




Fig. 55. 



portion. A differential action, therefore, ensues, and the 
effective value of the compensating current is less than it 
was before. 

In order to produce a constant potential on circuits 
having a variable inductance as well as a variable resist- 



78 



ALTERNATING-CURRENT MACHINES. 



ance, the General Electric Co. has designed its compensated 
revolving field generators, which are constructed for two- 
or three-phase circuits. The machine, Fig. 57, is of the 




Fig. 57. 

revolving field type, the field being wound with but one 
simple set of coils. On the same shaft as the field, and 
close beside it, is the armature of the exciter, as shown 
in Fig. 58. The outer casting contains the alternator 
armature windings, and close beside them the field of the 
exciter. This latter has as many poles as has the field of 
the alternator. Alternator and exciter, therefore, operate 
in a synchronous relation. The armature of the exciter is 
fitted with a regular commutator, which delivers direct 
current both to the exciter field and, through two slip- 



ALTERNATORS. 



79 



rings, to the alternator field. On the end of the shaft, 
outside of the bearings, is a set of slip-rings, four for a 
quart er-phaser, three for a three-phaser, through which 
the exciter armature receives alternating current from one 
or several series transformers inserted in the mains which 
lead from the alternator. This alternating current is 
passed through the exciter armature in such a manner as 
to cause an armature reaction, as described in § 36, that 
increases the magnetic flux. This raises the exciter vol- 
tage and hence increases the main field current. The 




Fig. 58. 

reactive magnetization produced in the exciter field is 
proportional to the magnitude and phase of the alternating 
current in the exciter armature. The reactive demag- 
netization of the alternator field is proportional to the 
magnitude and phase of the current in the alternator 
armature. And these currents have the fixed relations 
of current strength and phase, which are determined by the 
series transformers. Hence the exciter voltage varies so 
as to compensate for any drop in the terminal voltage. 
Neither the commutator nor any of the slip-rings carry 
pressures of over 75 volts. The amount of over-com- 



8o 



ALTERNATING-CURRENT MACHINES. 



pounding is determined by the ratio in the series trans- 
formers. The normal voltage of the alternator may be 
regulated by a small rheostat in the field circuit of the 
exciter. 

43. Inductor Alternators Generators in which both 

armature and field coils are stationary are called inductor 
alternators. Fig. 59 shows the principle of operation of 

/ARMATURE COILS , 




FIELD COIL 



^ 



Fig. 59. 

these machines. A moving member, carrying no wire, 
has pairs of soft iron projections, which are called induc- 
tors. These projections are magnetized by the current 
flowing in the annular field coil as shown in figure. The 
surrounding frame has internal projections corresponding 
to the inductors in number and size. These latter projec- 
tions constitute the cores of armature coils. When the 
faces of the inductors are directly opposite to the faces of 
the armature poles, the magnetic reluctance is a minimum, 
and the flux through the armature coil accordingly a maxi- 
mum. For the opposite reason, when the inductors are 
in an intermediate position the flux linked with the arm.a- 
ture coils is a minimum. As the inductors revolve, the 
linked flux changes from a maximum to a minimum, but it 
does not change in sign. 

Absence of moving wire and the consequent liability to 



ALTERNATORS. 



chafing of insulation, absence of collecting devices and 
their attendant brush friction, and increased facilities for 




Fig. 60. 

insulation are claimed as advantages for this type of ma- 
chine. By suitably disposing of the coils, inductor alter- 
nators may be wound 
for single- or poly- 
phase currents. 

The Stanley Elec- 
tric Manufacturing 
Company manufac- 
ture two-phase induc- 
tor alternators. A 
view of one of their 
machines is given in 
Fig. 61. Fig. 60, with the 

frame separated for inspection of the windings. In this 
picture the field coil is hanging loosely between the pairs 




82 



ALTERNATING-CURRENT MACHINES. 




of inductors. The theoretical operation of this machine is 
essentially that described above. All iron parts, both 
stationary and revolving, that are subjected to pulsations 
of magnetic flux, are made up of laminated iron. The 



ALTERNATORS. 



83 




Fig. 63a. 




Fig. 63b. 



84 



ALTERNATING-CURRENT MACHINES. 



large field coil is wound on a copper spool. Ordinarily 
when the field circuit of a large generator is broken, the 



I I l| 




Fig. 64. 



U 



E.M.F. of self-induction may rise to so high a value as 
to pierce the insulation. With this construction the cop- 
per spool acts as a short circuit around the decaying flux, 



ALTERNATORS. 



85 



and prevents high E.M.F.'?> of self-induction. Figs. 6i 
and 62 show the details of construction of a Stanley 
machine of a larger size than the one previously shown. 
Inductor alternators are also manufactured by Westing- 
house and Warren companies. The construction of the 
machine made by the latter company is shown in Fig. 6^a. 
There is but a single field coil, which fits into the recess in 
the back of the frame as shown. The armature coils sur- 
round the pole projections, and the flux through them is 
altered by the change of reluctance caused by the ro- 
tating inductor which carries no wire. The exciter is 
carried upon a platform which (Fig. 6'^b) forms part of 
the main frame and is driven by a belt from a pulley on 
the armature shaft. 

44. Revolving Field Alternators In this type of al- 
ternator, the armature windings are placed on the inside 
of the surrounding frame, and the field poles project radi- 




Fig. 65. 



ally from the rotating member. As was stated before, this 
type of construction is to be recommended in the case of 
large machines which are required to give either high 



86 ALTERNATING-CURRENT MACHINES. 

voltages or large currents. With the same peripheral 
velocity, there is more space for the armature coils ; the 




SCALE fi INCH EQmitS ONE EOOT^ 

Fig. 66. 



coils can be better ventilated, air being forced through 
ducts by the rotating field ; stationary coils can be more 
perfectly insulated than moving ones ; and the only cur- 



ALTERNATORS. 



87 



rents to be collected by brushes and collector rings are 
those necessary to excite the fields. 

Fig. 64 shows a General Electric 750 k. w. revolving 
field generator. The two collector rings for the field cur- 



Field Current 
Ampere Turns 




.4 .5 .6 ,7 .8 .9 
Output-Proportion of full load 
Fig. 67. 



rent are shown, and in Fig. 65 the edgewise method of 
winding the field coils is shown. The collector rings are 
of cast iron and the brushes a^e of carbon. Fig. 66 shows 
the details of construction of a 5,000 k. w. three-phase 



88 ALTERNATING-CURRENT MACHINES. 




Fig. 68. 



ALTERNATORS. 



89 



6,600-volt machine of this type as constructed for the 
Metropohtan Street Railway Co. of New York. This 
machine has 40 poles, runs at 75 r. p. m. at a peripheral 




Fig. 69. 

velocity of 3,900 feet per minute. This gives a frequency 
of 25. The air gap varies from five-sixteenths at the 
pole center to eleven-sixteenths at the tips. The short- 
circuit current at fall-load excitation is less than 800 am- 
peres per leg. The rated full-load current is slightly over 
300 amperes. The efficiency and no-load saturation curve 
is shown in Fig. 6"j . 

The Bullock Electric Mfg. Co. also make generators of 
this type. The method of placing armature coils is shown 
in Fig. 68. These coils, as shown, are wire wound, taped, 
insulated, and held in slots by maple-wood wedges. The 
field poles are fastened directly to a spider having a heavy 
rim. The pole pieces are of T-shaped steel punchings, 



90 ALTERNATING-CURRENT MACHINES. 




Fig. 70. 



held together by rivets and malleable iron end pieces. 
These are fastened to the rim of the spider by bolts in the 
case of slow-speed machines, or are dovetailed to fit slots 
in the rim in the case of high-speed machines. This latter 



ALTERNATORS. 9I 

method of fastening is shown in Fig. 69, which represents 
the field and shaft of a small-sized high-speed machine. 

The Westinghouse rotating field consists of a steel rim 
mounted upon a cast-iron spider. Into dovetailed slots in 
the rim are fitted laminated plates with staggered joints. 
These plates are bolted together. The laminations are 
supplied at intervals with ventilating ducts. The coils are 
kept in place by retaining wedges of non-magnetic material 
A portion of a field is shown in Fig. 70. 



92 



ALTERNATING-CURRENT MACHINES. 



CHAPTER VI 



THE TRANSFORMER. 



45. Definitions. — The alternating-current transformer 
consists of one magnetic circuit interlinked with two elec- 
tric circuits, of which one, the pinmary, receives electrical 
energy, and the other, the secondary, delivers electrical 
energy. If the electric circuits surround the magnetic 
circuit, as in Fig. 71, the transformer is said to be of the 

core type. If the re- 
verse is true, as in 
Fig. 72, the trans- 
former is of the sJiell 
type. The practical 
utility of the trans- 
former lies in the fact 
that, when suitably de- 
signed, its primary can 
take electric energy at 
one potential, and its 
secondary deliver the 
same energy at somie 
other potential ; the ratio of the current in the primary to 
that in the secondary being approximately inversely as the 
ratio of the pressure on the primary to that on the secon- 
dary. 

The ratio of transformation of a transformer is repre- 




Fig. 71- 



THE TRANSFORMER. 93 

sented by r, and is the ratio of the number of turns in the 
secondary coils to the number of turns in the primary coil. 
This would also be the ratio of the secondary voltage to 




Fig. 72. 

the primary voltage, if there were no losses in the trans- 
former. A transformer in which this ratio is greater than 
unity is called a '' step-up " transformer, since it delivers 
electrical energy at a higher pressure than that at which 
it is received. When the ratio is less than unity it is 
called a " step-down " transformer. Step-up transformers 
find their chief use in generating plants, where because of 
the practical limitations of alternators, the alternating cur- 
rent generated is not of as high a potential as is demanded 
for economical transmission. Step-down transformers find 
their greatest use at or near the points of consumption of 
energy, where the pressure is reduced to a degree suitable 
for the service it must perform. The conventional repre- 
sentation of a transformer is given in Fig. 73. In general, 
little or no effort is made to indicate the ratio of trans- 
formation by the relative number of angles or loops shown, 



94 ALTERNATING-CURRENT MACHINES. 

though the low-tension side is sometimes distinguished 
from the high-tension side by this means. 

When using tlie same or part of tlie same electric cir- 
cuit for both primary and secondary, the device is called 
an auto-transformer. These are sometimes used in the 




s^ 



Si 



ig ^ ^ 



Fig. 73- Fig. 74. 

Starting devices for induction motors, and sometimes 
connected in series in an alternating-current circuit, and 
arranged to vary the E.M.F. in that circuit. Fig. 74 is 
the conventional representation of an auto-transformer. 

46. Core Flux — (^7) Open-circuited secondary. \\'hen 
the secondary coil of a transformer is open-circuited it is 
perfectly idle, having no influence on the rest of the ap- 
paratus, and the primary becomes then merely a choke 
coil. A transformer is so designed that its reactance is 
very high, and its resistance comparatively low. This 
makes a large impedance, which is almost wholly reactive ; 
hence the current that will flow in the primary when the 
secondary is open-circuited is very small, and lags practi- 
cally 90° behind the E.M.F, which sends it. This 
current is called the excitino- current, or sometimes less 
properly the magnetizing current or leakage current. A 
flux is set up in the iron of a transformer, which is sinu- 
soidal and is in phase with the exciting current. This flux 
induces a practically sinusoidal E.M.F. in the primary 
coil, 90° behind it in phase ; because the induced E.M.F. 
is greatest when the time rate of flux change is greatest, 
and the flux changes fastest as it is passing through the 



THE TRANSFORMER. 95 

zero value. This induced E.M.F. is 90° behind the flux, 
whicii in turn is 90^" behind the impressed pressure; there- 
fore the induced E.M.F. is 180° behind the impressed 
E.M.F. or is a coimtei^ E.M.F. The counter pressure is 
less than the impressed pressure only by the small amount 
necessary to cause the small exciting current to flow. 
Neglecting the primary resistance, R^^, and the reluctance, 
(R, of the core, the counter pressure would be equal to the 
impressed pressure ; and in commercial transformers this 
is true to within a small percentage. Considering that 
the flux varies sinusoidally, and that its maximum value 
is $„^ ; then the flux at any time, t, is $^ cos w/, and the 
counter E.M.F., which is equal and opposed to the im- 
pressed primary pressure E^,, may be written (§ 13, vol. i.) 





^, 11^^ ^/(#,,COSa)/)^ 




^^' IO« dt ' 


and since ^,^ 


and (0 are constant 




Ep = I o~^ iip<^^,n sin (o/, 


from which 






E^^= io-^//^o)$,„, 


and 


n^^oi n 00 



This equation is used in designing transformers and 
choke coils. The values of $,^ for 60 cycle transformers 
of different capacities, as determined by experiment and 
use, are shown in the curve. Fig. 75. It is usual in such 
designs to also assume a maximum flux density, (B,,^. 
While the value assumed differs much with different man- 
ufacturers, it is safe to say that for 25 cycles (B^ varies 
between 9 and 1 4 kilogausses ; for 60 cycles between 6 
and 9 kilogausses; and for 125 cycles between 5 and 7 



96 



ALTERNATING-CURRENT ^lACHINES. 



kilogausses. The necessary cross-section, A, of iron, neces- 
sary to give the desired counter £.JI.K, as well as the 
number of turns of wire in the primary, is then found from 
the above, as 



<l> 



a,„^. 



(d) With secondary closed through an ontside impedance. 
The flux, which is linked with the primary, is also linked 
with the secondary. Its variations produce in the secon- 



o 





1 ^ j^ 


^,^^-^^^1 




1 ^^^^^ ! 






1 ^^ 








1 X 


Lighting Transformers 
at 60^ 








Y \ 1 ' 








/• ' i i 




\ 




/ i 1 1 1 i 






1 



10 12 14 16 13 20 22 

Capacity in Kilowatts 
Fig. 75. 



dary an E.JI.F. r times as great as the coimter E.M.F. in 
the primary, since there are r times as many turns in the 
secondary coil as in the primary, or 



tE, 



If this secondary be closed through an external impedance, 
a current I, will flow through this circuit. In the secon- 
dar}^ coil the ampere turns, ;//„ ^^ill be opposed to the 
ampere turns of the primary, and will thus tend to demag- 
netize the core. This tendency is opposed by a read- 
justment of the conditions in the primary circuit. Any 
demagnetization tends to lessen the counter E.M.F. in the 
primar}' coil, which immediately allows more current to 



THE TRANSFORMER. 9/ 

flow in the primary, and thus restores the magnetization to 
a value but shghtly less than the value on open-circuited 
secondary. Thus the core flux remains practically con- 
stant whether the secondary be loaded or not, the ampere 
turns of the secondary being opposed by a but slightly 
greater number of ampere turns in the primary. So 

71 J^ = nj^^, very nearly, 

11 I 

and 7, = -? T; = - t; . 

The counter E.M.F. in the primary of a transformer 
accommodates itself to variations of load on the secondary 
in a manner similar to the variation of the counter E.M.F. 
of a shunt wound motor under varying mechanical loads. 

If the secondary load be inductive or condensive, then /« 
will lag or lead E^ by the same angle that I^ lags or 
leads Epy still neglecting i?^„ R^, (R, and hysteresis. In 
such case 7^ is i8o^ from, or opposite to, /g, and E^, is oppo- 
site to E^. For a more exact statement than the above, 
see § 54. 

47. Equivalent Resistance and Reactance of a Trans- 
former. — ^ If a current of definite magnitude and lag be 
taken from the secondary of a transformer, a current of 
the same lag and r times that magnitude will flow in the 
primary, neglecting resistance, reluctance, and hysteresis. 
An impedance which, placed across the primary mains, 
would allow an exactly similar current to flow as this 
primary current, is called an eqitivalent impedance, and its 
components are called equivalent irsistajice and equivalent 
reactance. 

If the whole secondary circuit of a transformer with its 
load have a resistance R^ and a reactance X^, and if the 



98 ALTERNATING-CURRENT MACHINES. 

primary pressure be E^, and the secondary total pressure 
E,^ then the current that will flow in the secondary circuit is 

E. 



and it lags behind E^ by an angle 4>, whose tangent is — • 

^. 

Therefore V^TTX' = — ■ 

8 

If the equivalent impedance have a resistance R and a 

reactance X then the ratios ^— and ^' must be equal, since 

the angle of current lag is the same in both primary and 
secondary. And since the current in the equivalent im- 
pedance has the same magnitude as that in the primary 

E,. 



and 
But 
and 



therefore, Vi?^ + X^ = r7 = :72 "/ = zr2 ^^'' + ^''' 

But 

Solvino; 



which are the values of the equivalent resistance and re- 
actance respectively. 

*. K • 







I,- 


= rl,. 


E, 


-\^^' 


E. 

T 


I E. 


E 


R. 


X~ 


X. 


E = 


-\r. 

T- 


X = 


-7^-'^ 



THE TRANSFORMER. 99 

48. Transformer Losses. — The transformer as thus far 
discussed would have 100% efficiency, no power whatever 
being consumed in the apparatus. The efficiencies of 
loaded commercial transformers are very high, being gen- 
erally above 95% and frequently above 98%. The losses 
in the apparatus are due to (a) the resistance of the elec- 
tric circuits, (b) reluctance of the magnetic circuit, (c) 
hysteresis, and {d) eddy currents. These losses may be 
divided into core losses and copper losses, according as to 
whether they occur in the iron or the wire of the trans- 
former. 

49. Core Losses. — {a) Eddy curre7it loss. If the core 
of a transformer were made of solid iron, strong eddy cur- 
rents would be induced in it. These currents would not 
only cause excessive heating of the core, but would tend 
to demagnetize it, and would require excessive currents to 
flow in the primary winding in order to set up sufficient 
counter E.M.F. 

To a great extent these troubles are prevented by mak- 
ing the core of laminated iron, the laminae being trans- 
verse to the direction of flow of the eddy currents but 
longitudinal with the magnetic flux. Each lamina is more 
or less thoroughly insulated from its neighbors by the 
natural oxide on the surface or by Japan lacquer. The 
eddy current loss is practically independent of the load. 

The E.M.F. producing these eddy currents is in phase 
with the counter E.M.F. of the primary coil, both being 
produced by the same flux. Its value E^ is expressed by 

P 

the fraction--^, where P^ is the power loss in watts due 

to eddy currents, and T^ is the exciting or no-load primary 

current. The value of P^ is calculated from the following 
LofC. 



100 ALTERNATING-CL^RRENT MACHINES. 

empirical formula, in which perfect insulation between the 
laminae is assumed : 

where 

k = 2i constant depending upon the reluctivity and 

resistivity of the iron. 
V = volume of iron in cm.^, 
/ = thickness of one lamina in cm., 
/ = frequency, 
and 

(B„,= maximum flux density ($,^ per cm.-). 

In practice k has a value of about 1.6 x io~^\ 

[b) Hysteresis loss. A certain amount of power, P;,, 
due to the presence of hysteresis, is required to carry the 
iron through its cyclic changes. The value of P^ can be 
calculated from the formula expressing Steinmetz's Law, 

where 

V = volume of iron in cm.^, 

/ = frequency, 

(E,,.= the maximum flux density, 
and rj = the hysteretic constant (.002 to .003). 

The portion of the impressed E.M.F. which must be 
expended in the primary circuit to balance the hysteretic 
loss is 

•■- A 
This is in phase Avith /^. 

Closely associated with Ej^ is another portion of the 
impressed E.M.F. which is consumed in producing the 
cyclical and sinusoidal variations of magnetic Allx. This is 
not easily considered distinct from Ej^. Consider, however, 
the primary current. There is but one primary current. 



THE TRANSFORMER. lOI 

At any instant of time a portion of it is balanced and its 
magnetic effect is neutralized by the demagnetizing cur- 
rent in the secondary ; another portion is balanced by the 
demagnetizing action of the eddy currents ; and the rem- 
nant is useful in producing the cycHcal variations of the 
magnetic flux. If the flux be sinusoidal this portion of 
the current cannot be sinusoidal. This is due to the 
change in permeability with saturation of the iron core. 
Neither is the rising current curve the reverse of the fall- 
ing current curve. This is due to the fact that, owing to 
hysteresis, the permeability on rising flux is smaller than 
on falling under a given magnetomotive force. This last 
portion of the primary current is therefore not sinusoidal. 
As it is but a small percentage of the total current, it is, 
however, for convenience generally considered as sinusoi- 
dal. To send this distorted portion of the primary current 
requires a portion of the impressed E.M.F., and this is 
made up of two components, — E^^ in phase with the pri- 
mary current and discussed above, and E^^^ at right angles 
with the primary current. This E^^^^ may be considered as 
sending that portion of the current sufficient to overcome 
the magnetic reluctance of the core. Being at right angles 
with /^^ it represents no loss of power. During half of the 
time I^^ and E^^^^ have the same direction and during the 
other half they are in opposite directions. The core there- 
fore alternately receives energy from the circuit and gives 
it back to the circuit. 

To determine the value of ^,„„^ consider that it must be 
of such a magnitude as will send through the primary coil, 
of resistance R^^, that portion, 7,,^^^, of the main current 
which produces the flux, 

F = R r 



I02 ALTERNATING-CURRENT MACHINES. 

Representing the reluctance of the core by (R, and the 
magnetomotive force necessary to produce the flux $^ by 
5C, from §§ 21 and 25, vol. i., 







^ = 


3C '-"'':: 














cR 


(R ' 










c>T-> (-•/:» 




f...^ 


10 (R$ 

4 '^ ^^p 


1 (R$,^ 


> 








CliL^C 


4 V2 TT//^, 




I 




^.nag = 


4 V^T^^/;, 












\na, is 


called 


the magnetizing 


current 


of 


a 


transfor 


mer. 



The primary counter E.AI.F., E, is less than the primary 
line voltage by the slight pressure necessary to send this 
current through the primary resistance, thus, 

F — F — r R 

The value of (R is calculated (§ 24, vol. i.) from 

where / is the length of magnetic circuit, A its cross-sec- 
tion and p the reluctivity of the iron 

I I \ 



/x permeability/ 

In modern commercial transformers the core loss at 
60 ~ may be about 70% hysteresis and 30% eddy current 
loss. At 125 ~ it may be about 55% hysteresis and 45% 
eddy current loss. This might be expected, since it was 
shown that the first power of /enters into the formula for 
hysteresis loss, while the second power of/ enters into the 
formula for eddy current loss. 



THE TRANSFORMER. 103 

The core loss is also dependent upon the wave-form of 
the impressed E.M.F., a peaked wave giving a somewhat 
lower core loss than a flat wave. It is not uncommon to 
find alternators giving waves so peaked that transformers 
tested by current from them show from 5% to 10% less 
core loss than they would if tested by a true sine wave. 
On the other hand generators sometimes give waves so 
flat that the core loss will be greater than that obtained by 
the use of the sine wave. 

The magnitude of the core loss depends also upon the 
temperature of the iron. Both the hysteresis and eddy cur- 
rent losses decrease slightly as the temperature of the iron 
increases. In commercial transformers, a rise in tempera- 
ture of 40° C. will decrease the core loss from 5% to 10%. 
An accurate statement of the core loss thus requires that 
the conditions of temperature and wave-shape be specified. 

The core loss is practically constant at all loads, and is 
the same whether measured from the high-tension or the 
low-tension side, the exciting current in either case being 
the same percentage of the corresponding full-load current. 
The exciting current varies in magnitude with the design 
of the transformer. In general it will not exceed 5 % of 
the full-load current, and in standard lighting transformers 
it may be as low as 1%. In transformers designed with 
joints in the magnetic circuit the exciting current is largely 
influenced by the character of the joints, increasing if the 
joint is poorly constructed. In the measurement of core 
loss, if the product of the impressed volts by the exciting- 
current is less than twice the measured watts (i.e., if 
cos <^ >.5 or <^ < 60°) there is reason to suspect poorly 
constructed magnetic joints or higher densities in the iron 
than good practice allows. 



104 ALTERNATING-CURRENT MACHINES. 

50. Copper Losses. — The copper losses in a transformer 
are almost solely due to the regular current flowing 
through the coils. Eddy currents in the conductor are 
either negligible or considered together with the eddy cur- 
rents in the core. 

When the transformer has its secondary open-circuited 
the copper loss is merely that due to the exciting current 
in the primary coil, P^^g^v- This is very small, much 
smaller than the core loss, for both / „ and R^ are small 

' mag p 

quantities. When the transformer is regularly loaded the 
copper loss in watts may be expressed 

At full load this loss will considerably exceed the core 
loss. While the core loss is constant at all loads, the 
copper loss varies as the square of the load. 

51. Efficiency. — Since the efficiency of induction appa- 
ratus depends upon the wave-shape of E.M.F., it should be 
referred to a sine wave of E.M.F., except where expressly 
specified otherwise. The efficiency should be measured 
with non-inductive load, and at rated frequency, except 
where expressly specified otherwise. 

The efficiency of a transformer is expressed by the ratio 
of the net power output to the gross pow^er input or by 
the ratio of the power output to the power output plus all 
the losses. The efficiency, e, may then be written, 

' VJs+F,+ P, + F; 

where V^ is the difference of potential at the secondary 
terminals. 

If the transformer be artificially cooled, as many of the 



THE TRANSFORMER. I05 

larger ones are, then to this denominator must be added 
the power required by the cooHng device, as power con- 
sumed by the blower in air-blast transformers, and power 
consumed by the motor-driven pumps in oil or water 
cooled transformers. Where the same cooling apparatus 
supplies a number of transformers or is installed to supply 
future additions, allowance should be made therefor. 

Inasmuch as the losses in a transformer are affected by 
the temperature, the efficiency can be accurately specified 
only by reference to some definite temperature, such as 
25° C. 

The all-day efficiency of a transformer is the ratio of 
energy output to the energy input during the twenty-four 
hours. The usual conditions of practice will be met if the 
calculation is based on the assumption of five hours full- 
load and nineteen hours no-load in transformers used for 
ordinary lighting service. With a given limit to the first 
cost, the losses should be so adjusted as to give a maximum 
all-day efficiency. For instance, a transformer supplying 
a private residence with light will be loaded but a few 
hours each night. It should have relatively much copper 
and little iron. This will make the core losses, which con- 
tinue through the twenty-four hours, small, and the copper 
losses, which last but a few hours, comparatively large. 
Too much copper in a transformer, however, results in bad 
regulation. In the case of a transformer working all the 
time under load, there should be a greater proportion of 
iron, thus requiring less copper and giving less copper loss. 
This is desirable in that a loaded transformer has usually 
a much greater copper loss than core loss, and a halving 
of the former is profitably purchased even at the expense 
of doubling the latter. 



I06 ALTERNATING-CURRENT MACHINES. 

52. Regulation The definition of the regulation of a 

transformer as authorized by the American Institute of 
Electrical Engineers is as follows : ''In transformers the 
regulation is the ratio of the rise of secondary terminal 
voltage from full-load to no-load (at constant impressed 
primary terminal voltage) to the secondary full-load volt- 
age." Further conditions are that the frequency be kept 
constant, that the wave of impressed E.M.F. be sinusoidal, 
and that the load be non-inductive. 

Not the whole primary impressed pressure is operative 
in producing secondary pressure, for I^ Rp volts are lost in 
overcoming the resistance of the primary coil. Besides 
this there is a flux hnked with the primary that does not 
link the secondary. This induces a counter pressure in 
the primary which neutralizes a part of the impressed 
pressure. Such flux, linking one coil but not the other, is 
called leakage flux. Furthermore, not all of the E.M.F. 
induced in the secondary is utilizable at the terminals. 
There is a drop of /^ R^ volts due to the resistance of the 
secondary coil, and another drop due to a leakage flux 
which links the secondary but not the primary. All these 
drops increase with load, and therefore, neglecting core 
loss effects, at no load E^ = tEj„ but on load, E^<^rE^,, and 
the percentage of the full-load secondary pressure repre- 
sented by this fall is the regulation. 

The leakage flux affects the action of a transformer just 
the same as would an inductance connected in series with 
the same transformer, the latter having no leakage flux. 

The leakage flux increases with the current ; and if, for a 
current / it be $, then the value of the inductance, Z, is 



THE TRANSFORMER. 10/ 

where n is the number of turns in the coil. A method of 
calculating L, the equivalent indiLctance, is given in the 
next article. 

The resistance of the secondary causes a drop /^ R^. 
The same effect on the regulation would be caused if the 
secondary resistance were zero and another resistance 

whose value is R, = -^ were inserted in the primary cir- 
cuit. The imaginary primary drop, resulting from this 
insertion, has to be but - as great as the actual secondary 

T 

drop to be as great a percentage of the impressed E, 
and there is r times as much current to cause it, hence 

i?g = -^' . The power lost in this imaginary resistance is 

7/7^3, and this equals the power really lost in the secondary 

I^R,, since 7^, = t7, and R,= ^ . 

t" 

In order to calculate the regulation, consider this equiv- 
alent of secondary drop to be accounted for in the primary. 
Then for a given impressed E.M.F. on the primary, E^,, the 
terminal voltage on the secondary will be 
at no load e = rE 

at primary load I^,, 

K=r \_E^ - IXR, + R^ cos <^ - o^LI^, sin c/>], 

where E^ = secondary pressure generated, 

V^ = difference of potential at secondary terminals, 
Z = Zj,-{- Zg as calculated in the next section, 

and <^ = angle of lag of /^ behind E^,. 

Then from the definition of regulation, when 7 in the 
above is made equal to the full-load current, 

tE — V, 
Regulation = ~ — ^ » 



io8 



ALTERNATING-CURRENT MACHINES. 



53. Calculation of Equivalent Leakage Inductance. — 

The arrangement of one of the most usual kinds of core 

type transformers called 
the ''type H," is shown 
in Fig. y6. The coarse 
wire is wound inside the 
fine wire, and as these 
are more generally used 
as step-down transformers 
the latter will be called 
the primary. 

Fig. yy shows one leg 
of the transformer, giving 
the paths of leakage flux 
and the system of nota- 
tion employed. The discussion is carried on entirely in 
c. G. s. units. Consider the secondary (coarse wire) coil 
first. 




Fig. 76. 



1 




li 


G 




1!^^ 




g 


^ 


^ 






' 


f.-Y-> 


i<-x->; 



>. 


x-~ 


\ / 


1 
1 


1 


i« 


£^1 






! C 


s| 










1 
1 

1 




; \^ 




"~ 





Primary. 




£ 


r 


Secqndary^_^ 

i; 




1^ 


:■ B : 


|i 
1 
|l 

i| 
II 


!l 




II 











Fig. 77- 

The J/.Jl.F. tending to send flux through the elemen- 

v 
tary portion d.v and back through the iron is -^ of the 

whole J/.Jl.F. of the secondary, so for any element, 

X 



M.M.F. = 4 TT//,/ 



X 



THE TRANSFORMER. IO9 

Since the permeability of iron is roughly 1000 times 
that of air, no appreciable error is introduced by consider- 
ing the whole reluctance of the circuit of the leakage flux 
to be in the air portion of that circuit. If it be assumed 
that the lines of force follow a circular path from the end 
of the coil to the iron, the length of the air portion of the 
magnetic circuit for any element is C -\- ttx. The use of 
this value will result in an integral expression, simple 
enough in theory, but too unwieldy to be introduced on 
these pages. Since the portion of the air path outside 
the coil (the curved portion) is a small part of the whole 
path, no serious error will be introduced by assuming 
that the leakage flux from any element follows a path 

whose length is the average length C + -n- — • The cross- 

2 

section area of the air part of the magnetic circuit for any 
element is {2A-\-2B-\-Sx)dx. Therefore the reluctance 
of any element is 

C + '^X 



2 (^A -\- B -\- ^x) dx 
The elementary leakage flux, d^, is then 



M.M.F. ^tviij.x 2 (A -\- B -\- /\. x) dx 



C+ -X 



2 

X 



Since this flux links with — of the secondary turns, the 

X 

number of linka^'es is 



8 -KTiJ^ (A -\-B -}- 4 x) xdx XII ^ _ 8 TT ;// i^{A -\- B -^ \ x^ x^dx 



no ALTERNATING-CURRENT MACHINES. 

By definition (§ 8) the coefficient of self-induction, /, 
is numerically equal to the number of linkages per unit 
current. Therefore 

linkages 8 -rrJi^iA -\- B -\- 4.x) x^dx 
The limits of the variable x are o and A\ therefore 



1 = 



C-{-~X]X' 



c + ^xlx^ 



(A + B) jx'dx + 4 jx'dxl 



^X^ + X'„ 



/ = 8...^X^+^ + 3^ 



2 



This applies to one leg of the transformer. For the two 
legs, upon reverting to practical units, 

^6 ..2x.^+^ + 3A^ 



Z^= ^-^-trll'-X 



all the terms of which are either absolute numbers or 
linear dimensions in centimeters. 

It cannot be objected that this analysis does not take 
account of the leakage flux that does not travel the whole 
length of the coil, C. It is a true statement for any 
length, and therefore might be applied to the elementary 
length dC, which when integrated would give the result 
stated above. 

The value of L is determined in the same way, and the 



THE TRANSFORMER. Ill 

expression therefore is quite similar. There can be no 
iron in the path of the leakage flux from the outside coil, 
so the reluctance will be twice as great. The value that is 
represented by A for the inner coil becomes A + 2X -{- 2g 
for the outer. Likewise B is replaced by ^ + 2X -\- 2g, g 
being the space occupied by insulation between the coils. 
Then 

^,- ^^,^'h ^ 3(C + 7rF) 

If the secondary circuit is open the secondary coil is 
idle, equivalent to so much air, and all the flux set up by 
the primary is leakage flux. 

As the secondary resistance can be replaced by an 

equivalent primary resistance, R^ = -^ , for purposes of 
calculation, so also the secondary inductance can be re- 
placed by an equivalent inductance in the primary, L^ = — '. 

These values, L^, and L^, are to be used in the formula at 
the end of the last section for determining the regulation 
of a transformer. 

54. Exact Solution of a Transformer In the treat- 
ment of regulation, efficiency, etc., heretofore, certain 
small errors have been allowed, due to neglecting the 
effects of the core, eddy currents, and hysteresis losses. 
The following graphic solution, adapted from Steinmetz, 
takes account of all these effects, and is general in all 
respects. 

It must first be understood that there are three fluxes 
to be considered : (i) The useful flux that links both coils. 
It is not in any definite phase with either l^^ or /,. It is, 
however, always at right angles to the E.M.F. it induces, 



112 ALTERXATIXG-CURREXT MACHINES. 

the direct in the secondary, and the counter in the primary. 
(2) The leakage flux of the primary coil. This links the 
primary only, and being independent of /^ is always in 
phase ^Yith I^. ( 3 ) The leakage flux of the secondary coil. 
This is similarly in phase with 1^. 

Let Es^ E.M.F. induced in secondar}-. 

V\ = ditlerence of potential at secondar}- ter- 
minals. 
£^, = impressed primaiy pressure. 

/ -E 

E^ = operative part of E^^ (E,^ =^ 

I^ and I, = primaiy and secondaiy currents respectively, 
(f}^ and (/), = lag of primaiy and of secondaiy currents 
respectively behind E_^. and E,. 
y =z angle of lag of J/.J/.E behind useful flux. 

The problem is : Given the necessary data of the trans- 
former, to determine its behavior with any specified load 
on the secondary. 

As in Fig. 78, draw the line $, representing the direc- 
tion of the flux, vertically for convenience. In this analy- 
sis, the no-load exciting current is separated into two com- 
ponents. One is used in neutralizing the demagnetizing 
effect of the eddy currents. The other, /;.,, is the magne- 
tizing current and is also made up of two components, one 
in phase with the primar}" pressure E^^ and the other at 
right angles with it. The relative magnitudes of these 
two components are dependent upon the shape of the 
hysteresis curve of the iron. Once determined they may 
be represented as Ij, cos ^ and Ij.^ sin (3 where y8 is termed 
the angle of hysteretic lag. AA'hen multiplied by E^ the 
first represents the power lost in hysteresis ; the second 
the power passing backward and forward between the 



THE TRANSFORMER. 



113 



magnetic field and the circuit. If to the former the power 
lost in eddy currents, We, be added and the two be com- 
bined with the latter as in Fig. 79 an angle y results, which 




Fig. 78. 

represents the lag of the magnetomotive force. Determine 
the angle y in this manner. Draw the line M.M.F. (Fig. 
78) y ahead of $ indicating in direction and magnitude the 
ampere turns which must exist to set up 
the flux $. Its value is determined dur- 
ing the transformer design. Draw from 
the center the line Bo, 90° ahead of the 
flux, representing the operative primary 

pressure. Its length is - Bg and as it 

T 

Opposes the counter primary pressure, it is 
set ahead of the flux. Draw the line E^, 
90° behind $, representing the pressure induced in the 
secondary. Its length is proportional to the no-load sec- 
ondary terminal pressure. 




We I EpI;, cos/3 
Fig. 79. 



114 ALTERXATIXG-CURREXT MACHINES. 

The angle <^,, the lag due to the whole secondary cir- 
cuit, is knoAvn. Draw I^ at (^, behind E^, and extend it 
till its length is proportional to the secondary ampere 
turns, /^ 11^. This line represents one component of the 
magnetizing force. From this component line and the 
resultant line M.M.F. determine the other component 
/^ 11^^. Divide this by /r and the primary current is dis- 
covered in magnitude and phase. 

There is a drop of I^, R^, volts in the primary. The im- 
pressed pressure that compensates for this is in phase with 
/. A counter voltage 90° behind I^ will be set up due to 
the primary leakage flux. Its value is wZ^, I^,. To over- 
come this an impressed pressure must be supplied opposite 
it in phase or 90° ahead of the current. In a side figure 
vectorially add I^,Rp in the phase of I^^ and ^L^,I^^ 90° 
ahead of this phase. This gives the dii'ection and magmi- 
tude of the drop d^, in the primar}-. Properly add d^ to the 
operative pressure E^^ and the necessary impressed pressure 
E^^ is the resultant. The angle between /^^ and E^^ is 
the angle of lag <^^, of the primary current. It slightly 
exceeds f^^. 

The pressure Es is generated in the secondary coil. 
There is a drop of /^ R^ volts in this coil in phase with /^. 
A counter voltage 90" behind /, will be set up due to the 
secondary leakage flux. Its value is wZ^ /^. To overcome 
this wZ^ I^ volts generated at 180° from this (i.e., 90° ahead 
of ZJ will be consumed. In a side figure vectorially add 
Z^ R^ in the phase of Z^ and ^L^I^ at 90° ahead of this 
phase. This gives the drop d^ in the secondary coil. This 
drop must be subtracted from the pressure generated to 
give the secondary terminal volts. To subtract a vector, 
revolve it 180° and proceed as in addition. Properly sub- 



THE TRANSFORMER. 



115 



tract d^ from E^ and the resultant, f^, is the potential dif- 
ference at the terminals of the secondary coil of the 
transformer. 

By constructing this diagram for full load I^ and then 
for /, = o, the regulation of a transformer can be found by 
the ratio of the difference between the values of V^ in each 
case to the full load V^. The efficiency at any load can be 
determined from the diagram for that load, by 

EJ^ cos <^5 
^ ~ EpI,, cos c^^, ' 

Fig. 78 is not the true diagram of a commercial trans- 
former. For clearness a ratio of i to i has been portrayed 
and the losses greatly exaggerated. In practice it will be 
found impossible to complete the solution graphically 
because of the extreme flatness of the triangles. The 
better way is to draw an exaggerated but clear diagram, 
and obtain the true values of the sides by the algebra of 
complex imaginary quantities, or if the student is unfa- 
miliar with this method, by the more laborious methods of 
trigonometry and geometry. 

55. Methods of Connecting Transformers There are 

numerous methods of connecting transformers to distribut- 
ing circuits. The simplest case is 
that of a single transformer in a 
single-phase circuit. Fig. 80 shows 
such an arrangement. This and 
the succeeding figures have the 
pressure and current values of the 
different parts marked on them, as- 
suming in each case a i k.w., i to 10 ^^^' ^°' 
step-down transformer. As in Fig. 81, two or more trans- 




ii6 



ALTERNATING-CURRENT MACHINES. 



formers may have their primaries in parallel on the same 
circuit, and have their secondaries independent. If the two 
secondaries of this case are connected properly in series 
a secondary system of double the potential will result, or 
by adding a third wire to the point of juncture, as shown 
by the dotted line of Fig. 82, a three-wire system of dis- 
tribution can be secured. The secondaries must be con- 
nected cumulatively ; that is, their instantaneous E.M.F.'s 
must be in the same direction. If connected differentially, 
there would be no pressure between the two outside sec- 




Fig. 81. Fig. 82. 

ondary wires, the instantaneous pressures of the two coils 
being equal and opposed throughout the cycle. Again, 
with the same condition of primaries, the secondaries can 
be connected in multiple as in Fig. 83. Here the connec- 
tions must be such that at any instant the E.M.K's of the 
secondaries are toward the same distributing wire. The 
connection of more than two secondaries in series is not 
common, but where a complex network of secondary dis- 
tributing mains is fed at various points from a high-tension 
system, secondaries are necessarily put in multiple. 

In many types of modern transformers it is usual to 



THE TRANSFORMER. 



117 




Fig. 83. 



wind the secondaries (low-tension) in two separate and 
similar coils, all four ends being brought outside of the 
case. This allows of connections to two-wire systems of 
either of two pressures, or for a three-wire system accord- 
ing to Figs. 82 and 83, to be 
made with the one transformer, 
this being more economical than 
using two transformers of half 
the size, both in first cost and 
in cost of operation. In many 
transformers the primary coils 
are also wound in two parts. 
In these, however, the four ter- 
minals are not always brought 
outside, but in some cases are 
led to a porcelain block on which are four screw-connectors 
and a pair of brass links, allowing the coils to be arranged 
in series or in multiple according to the pressure of the 
line to which they are to be connected. From this block 

two wires run through suitably 
bushed holes outside the case. 
A two-phase four-wire system 
can be considered as two inde- 
pendent single-phase systems, 
transformation being accom- 
plished by putting similar single- 
Fig- 84. phase transformers in the circuit, 

one on each phase. If it is desired to tap a two-phase 
circuit to supply a two-phase three-wire circuit, the 
arrangement of Fig. 84 is employed. By the reverse 
connections two-phase three-wire can be transformed to 
two-phase four-wire. An interesting transformer connec- 



1 A 

r \ r\ '<lk 

1 A. 



i8 



ALTERNATING-CURRENT AIACHINES. 



tion is that devised by Scott, which permits of transfor- 
mation from two-phase four-wire to three-phase tliree-Avire. 
Fig. 85 shows the connections of the two transformers. 
If one of the transformers has a ratio of 10 to i with a 
tap at the middle point of its secondary coil, the other 



must have a ratio of 10 to .86/ (10 to — ^ J. One ter- 
minal of the secondary of the latter is connected to the 



t << 




1 ' 


^ < 




:> 








8 












I < 




i 








' t ^ 




> <T 


^ 




:> ' 




> < 


><'i 


s 












s. < 


r^g 


1 




^ 


> ^ ')' 






Fig. 85. 

middle of the former, the remaining three free terminals 
being connected respectively to the three-phase wires. In 
Fig. 86, considering the secondary coils only, let mn rep- 
resent the pressure generated in the first transformer. 
The pressure in the second transformer is at right angles 
(§ 5) to that in the first, and because of the manner of 
connection, proceeds from the center of viii. Therefore 
the hue op represents in position, direction, and magnitude 
the pressure generated in the second. From the geo- 
metric conditions mnp is an equilaterial triangle, and the 
pressures represented by the three sides are equal and at 
60° with the others. This is suitable for supplying a 
three-phase system. In power transmission plants it is 
not uncommon to find the generators wound two-phase, 
and the step-up transformers arranged to feed a three- 
phase line. 



THE TRANSFORMER. 



19 




In America it is common to use one transformer for 
each phase of a three-phase circuit. The three transform- 
ers may be connected either Y or A. They may be Y on 
the primary and A on the secondary, or vice versa. Fig. 
%^ shows both primary and 
secondary connected A. 
The pressure on each pri- 
mary is 1000 volts, and as 
a i-K.w. transformer was 
assumed, i.e., i k. w. per 
phase, there will be one 
ampere in each, calling for 
1-7 (^^3) amperes in each Fig. 87. 

primary main (§ 33). This arrangement is most desirable 
where continuity of service is requisite, for one of the 
transformers may be cut out and the system still be 
operative, the remaining transformers each taking up the 
difference between i and i the full load ; that is, if the 

system was running at 
full load, and one trans- 
former was cut out, the 
other two would be over- 
loaded i6| per cent. 
Even if two of them 
were cut out, service 
over the remaining phase 
could be maintained. It 

Fig. 88. 




ularly supply motors from three-phase mains by two some- 
what larger transformers rather than by three smaller 
ones. Fig. 88 shows the connections for both primaries 
and secondaries in Y- If hi this arrangement one trans- 



120 



ALTERNATING-CURRENT MACHINES. 



former be cut out, one wire of the system becomes idle, 
and only a reduced pressure can be maintained on the re- 
maining phase. The advantage of the star connection lies 
in the fact that each transformer need be wound for only 
57.7 per cent of the line voltage. In high-tension trans- 
mission this admits of building the transformers much 
smaller than would be necessary if they were A connected. 
Fig. 89 shows the connections for primaries in A, second- 
aries in Y ; and Fig. 90 those for primaries in Y and sec- 
ondaries in A. By taking advantage of these last two 
arrangements, it is possible to raise or lower the voltage 





Fig. 89. 



Fig. 90. 



with I to I transformers. With three i to i transform- 
ers, arranged as in Fig. 89, 100 volts can be transformed 
to 173 volts; while if connected as in Fig. 90, 100 volts 
can be transform.ed into 58 volts. 

Fig. 91 shows a transformer and another one connected 
as an autotransformer doing the same work. Since the 
required ratio of transformation is i to 2, the autotrans- 
former does the work of the regular transformer with one- 
half the first cost, one-half the losses, and one-half the 
drop in potential (regulation). The only objection to this 
method of transformation is that the primary and second- 



THE TRANSFORMER. 



121 



ary circuits are not separate. With the circuits grounded 
at certain points, there is danger that the insulation of the 
low-tension circuit may be subjected to the voltage of the 
high-tension circuit. One coil of an autotransformer must 
be wound for the lower voltage, and the other coil for the 



1000 V. 

Wnaaaa/vx?* 



One 100 Kw 
Transformer 
Ratio 1 to 2 



1000 V. ^ ^ -„ ,/ 



Losses not considered 



I 10 



000 V, 



Transformer 
q! Ratio 1 to 1 



—2000- 

Losses not considered 



Fig. 91. 



difference between the two voltages of transformation. 
The capacity of an autotransformer is found by multiply- 
ing the high-tension current by the difference between the 
two operative voltages. Autotransformers are often called 
compensators. Compensators are advantageously used 




<-1000-V.-> <1000 •v> 



— 2000-V.— 

Losses not considered 



SOOO-Vr* £^2000-V-H> 



-2000-V.- 



Three 16.5 Kv 
Transformers 
Ratio 1 to 1 



Losses not considere 



Fig. 92. 



where it is desired to raise the potential by a small 
amount, as in boosting pressure for very long feeders. 
Fig. 92 shows three i to 2 transformers connected in A on 
a three-phase system, and three i to i compensators con- 
nected in Y to do the same work. 

From a two-phase circuit, a single-phase E.M.F. of any 



122 ALTERXATIXG-CURREXT MACHINES. 

desired magnitude and any desired phase-angle may be 
secured by means of suitable transformers, as shown in 
I^ig- 93- Suppose the two phases Xand l^of a two-phase 
system be of lOO volts pressure, and it is desired to obtain 
a single-phase E.M.F. of looo volts and leading the phase 
X by 30°. As in Fig. 94, draw a line representing the 



PHASE X.o 





PHASE Y.^ _ ^ ^ 

DIRECTION OF PHASE X. 

Fig. 93- Fig. 94. 

direction of phase X. At right angles thereto, draw a line 
representing the direction of phase F. From their inter- 
section draw a line 1000 units long, making an angle of 
30° with X. It represents in direction and in length the 
phase and the pressure of the required E.M.F. Resolve 
this line into components along X and Y, and it becomes 
evident that the secondary of the transformer connected 
to X must supply the secondary circuit with "666 volts 
and that the secondary of the other must supply 500 volts. 
Therefore the transformer connected to X must step-up 
I to ^.66 and that connected to Fmust step-up i to 5. If 
10 amperes be the full load on the secondary circuit, the 
first transformer must have a capacity of S.66 k.w., and the 
second a capacity of 5 k.w. The load on A' and I^is not 
balanced. 

56. Lighting Transformers. — Because of their extensive 
use on lighting distributing systems, the various manufac- 
turers have to a o-reat extent standardized their lines of 
lighting transformers. Power transformers are not as yet 



THE TRANSFORMER. 



123 



well standardized, probably because they are generally used 
in such large units as to warrant a special design for 
each case. 

The Wagner Electric Mfg. Co.'s ^^ype M " transformer 
is illustrated in Fig. 95. It is of the shell type of con- 
struction, makers using this type claiming for it superiority 
of regulation and cool running. In the shell type the iron 




Fig. 95. 

is cooler than the rest of the transformer, in the core type 
it is hotter. As the "ageing " of the iron, or the increase 
of hysteretic coefficient with time, is believed to be aggra- 
vated by heat, this is claimed as a point of superiority of 
the shell type. However, the prime object in keeping a 
transformer cool is not to save the iron, but to protect the 
insulation ; and as the core type has less iron and generally 
less iron loss, the advantages do not seem to be remarkably 



124 



ALTERNATING-CURRENT MACHINES. 



in favor of either. In the Wagner ''type M " transformers 

the usual practice of having two sets of primaries and sec- 
ondaries is followed. 
Fig. 96 shows the three 
coils composing one 
set. A low-tension 
coil is situated between 
two high-tensioned 
coils, this arrangement 
being conducive to 
good regulation. The 
ideal method would be 
to have the coils still 
more subdivided and 
interspersed, but prac- 
tical reasons prohibit 
this. Fig. 97 shows 
■^^^' ^^* the arrangement of the 

coils in the shell. The space between the coils and the 

iron is left to facilitate the circulation of the oil in which 

they are submerged. 

The laminae for the 

shell are stamped 

each in two parts and 

assembled with joints 

staggered. As can be 

seen from the first 

cut, all the terminals 

of the tw^o primary 

and the two secon- 
dary coils are brought outside the case. The smaller sizes 

of this line of transformers, those under 1.5 k.w., have 






Fig. 97. 



THE TRANSFORMER. 



125 



sufficient area to allow their running without oil, so the 
manufacturers are enabled to fill the retaining case with 
an insulating compound which hardens on cooling. 

The General Electric Co.'s " H " transformers are of 
the core type. In Fig. "j^ was shown a sectional view giv- 
ing a good idea of the arrangement of parts in this type. 
Fig. 71 is also one of this line of transformers. In it is 
shown the tablet board of porcelain on which the connec- 
tions of the two high-tension coils may be changed from 
series to parallel or vice versa, 
so that only two high-tension 
wires are brought through the 
case. Fig. 98 shows the ar- 
rangement of the various parts 
in the assembled apparatus. 
The makers claim for this type 
that the coils run cooler because 
of their being more thoroughly 
surrounded with oil than those 
of the shell type. Another 
point brought forward is that 
copper is a better conductor of 
heat than iron ; the heat from the inner portions of the 
apparatus is more readily dissipated than in the shell type. 
The core has the advantage of being made up of simple 
rectangular punchings, and the disadvantage of having four 
instead of two joints in the magnetic circuit. A particular 
advantage of the "type H " transformer is the ease and 
certainty with which the primary windings can be sepa- 
rated from the secondary windings. A properly formed 
seamless cyhnder of fiber can be slipped over the inner 
winding and the outer one wound over it. This is much 




Fig. 98. 



126 



ALTERXATIXG-CURRENT MACHIXES. 



more secure than tape or other material that has to be 
wound on the coils. 




Fig. 99. 

The Westuighouse '* O. D." transformers are of the 
shell t}'pe. The construction of the separate parts is 
shown in Fig. 99. The coils are wound narrow and to the 
full depth, and high-tension and 
low-tension coils alternate side 
by side instead of from the 
center out. Fig. 100 shows a 
2 K.w. O. D. transformer ^Aith- 
out the case. A tablet board 
is used for the terminals of 
the high-tension coils, but the 
low-tension Awes are all run 
out of the case. Fig. loi 
shows one of the coils. TA"]De 
O. D. transformers are built 
from i to 25 K.w. for lighting 
and to 50 K.w. for power. 
Those of 10 K.w. or less are p^g_ ^^^ 




THE TRANSFORMER. 



1Z*J 




Fig. loi. 



in cast-iron cases, those above lo k.w. in corrugated iron 
cases with cast tops and bottoms. The corrugations quite 




Fig. 102. 

materially increase the radiating surface. The windings 
are submerged in oil. 

An example of the Stanley Electric Manufacturing Co.'s 



128 



ALTERXATIXG-CURREXT AIACHIXES. 



Standard line of "type A. O." transformers is given in 
Fig. 1 02. These are also of the shell type, \\-ith divided 
primaries and secondaries, four of the eight which belong 
to a sinsfle transformer beinsr shown in Fisr. 10^. 




Fig. 103. 



57. Cooling of Transformers The use of oil to assist 

in the dissipation of the heat produced during the opera- 
tions of transformers is almost universal in sizes of less 
than about 100 k.w., especially if designed for outdoor 
use. Some small transformers are designed to be self- 
ventilating, taking air in at the bottom, which goes out at 
top as a result of being heated. They are not well pro- 
tected from the weather, and are liable to have the natural 
draft cut off by the building of insects' nests. Larger 
transformers that are air cooled and that supply their own 
draft are used to some extent in central stations and other 
places where they can be properly protected and attended 
to. A forced draft is, however, the more common. Where 
such transformers are employed, there are usually a number 



THE TRANSFORMER. 



29 



of them ; and they are all set up over a large chamber into 
which air is forced by a blower, as indicated in Fig. 104. 

^k Qt ^^ 



^ 1 


/ 




\ / 


n 


\ / 


1^ 




/O' AIR CHAMBER 

i 1 




..XS....--^XS.N 


.x\N^i^N\\v^:i^s 


vXVxXV 


< ^x\-^\X<Sxx\^\\\X\-^N-^^ 



Fig. 104. 

Dampers regulate the flow of air through the transformers. 
They can be adjusted so that each transformer gets its 
proper share. 

Fig. 105 shows a General Electric Company's air-blast 
transformer in process of construction. The iron core is 
built up with spaces between the laminae at intervals ; and 
the coils, which are wound very thin, are assembled in 
small intermixed groups with air spaces maintained by 
pieces of insulation between them. The assembled struc- 
ture is subjected to heavy pressure, and is bound together 
to prevent the possibility of vibration in the coils due to 
the periodic tendency to repulsion between the primary and 
the secondary. These transformers are made in sizes from 
100 K.w. to 1000 K.w. and for pressures up to 35,000 volts. 

Another method of cooling a large oil transformer is to 
circulate the oil by means of a pump, passing it through a 
radiator where it can dissipate its heat. Again cold water 
is forced through coils of pipe in the transformer case, and 
it takes up the heat from the oil. There is the slight dan- 
ger in this method that the pipes may leak and the water 
may injure the insulation. Water-cooled transformers 
have been built up to 2000 k.w. capacity. 



I30 



ALTERNATING-CURRENT MACHINES. 



In those cases where the transformer requires some 
outside power for tl '^''^ration of a blower or a pump, 
the power thus used must be charged against the trans- 




Fig. 105. 

former when calculating its efficiency. In general this 
power will be considerably less than i % of the trans- 
former capacity. 

58. Constant-Current Transformers. — For operating 
series arc-light circuits from constant potential alternating- 
current mains, a device called a constant-current trans- 
former is frequently employed. A sketch showing the 
principle of operation is given in Fig. 106. A primary 
coil is fixed relative to the core, while a secondary coil is 



THE TRANSFORMER. 



131 



allowed room to move from a close contact with the 
primary to a considerable distance from it. This secon- 
dary coil is nearly but not entirely 
counter-balanced. If no current 
is taken off the secondary that 
coil rests upon the primary. 
When, however, a current flows 
in the two coils there is a repul- 
sion between them. The counter- 
poise is so adjusted that there is 
an equilibrium when the current 
is at the proper value. 




Fig. 106. 

If the current rises above this 
value the coil moves farther away, and there is an increased 



amount of leakage flux. 



This lowers the E.M.F. induced 




Fig. 107. 



132 



ALTERNATING-CURRENT MACHINES. 



in the secondary, and the current faHs to its normal value. 
Thus the transformer automatically delivers a constant 
current from its secondary when a constant potential is 
impressed on its primary. 

Fig. 107 shows the mechanism of such an apparatus as 
made by the General Electric Company. The cut is self- 
explanatory. Care is taken to have the leads to the mov- 




Fig. 108. 

ing coil very flexible. Transformers for 50 lamps or 
more are made with two sets of coils, one primary coil 
being at the bottom, the other at the top. The moving 
coils are balanced one against the other, avoiding the 
necessity of a very heavy counterweight. Fig. 108 shows 
a 50-light constant-current transformer without its case. 
Fig. 109 shows a complete 2 5 -lamp apparatus. The tank 



THE TRANSFORMER. 133 

is filled with oil, the same as an ordinary transformer. 
Great care must be taken to keep these transformers level, 
and to assist in this the larger sizes have spirit-levels built 




Fig. 109. 

into the case. A pair of these transformers can be spe- 
cially wound and connected to supply a series arc-light 
circuit from a three-phase line, keeping a balanced load on 
the latter. 

59. Design of a Transformer The method of design- 
ing a transformer depends upon the specifications as to con- 
struction and operation, and upon various values which 
the designer is forced or sees fit to assume. The following 
is one method : — 

Specificatio7is. — These usually give the capacity in 
watts, the frequency, the primary voltage, the secondary 



134 



ALTERXATIXG-CURREXT MACHINES. 



voltage, and the conditions of operation, place of installa- 
tion, whether loaded all day or not, etc. 

Assumptions. — The assumption of the following quanti- 
ties is usually preliminary to any calculation, — the shape 
of transformer, -the current density in the primary, the 
current density in the secondary, the turns in the primary 
coil, and the maximum flux density in the iron. The 
method of design is one of cut and try. A number of 
values of flux density and various numbers of primary turns 
are assumed. Efficiency curves are calculated for the 
various arrangements. The most efficient is ultimately 
selected ; or if none are satisfactory, the course of the 
design will have brought out the proper direction to take 
in making new assumptions. 

The following design refers to a core-type, step-down, 
lighting transformer of about 5 k. w. capacity. The as- 
sumptions are: 1000 circular mils per ampere in the 
primary, 1500 circular mils per ampere in the secondary 
(because this is inside, and has less opportunity of dissi- 

k— b— > 





Fig. no. 

pating its heat), 500, 700, and 1000 turns primary 
successively, and 2000, 3000, and 4000 gausses maximum 
flux density. The transformer will have the shape shown 
in Fig. no. Because of the general use of the English 
units of measure by most practical mechanics, the dimen- 



THE TRANSFORMER. 135 

sions indicated are all expressed in inches. The ratio 
- = m may be conveniently assumed as m = 1.5, and the 

ratio — = ;^ is likewise generally made n = i. 

I. To obtain the ai^ea, A, of the core in sqitare ce7iti- 
meters. 

Let E = impressed primary E.M.F.^ 

(B,„= assumed maximum flux density, 
7J, = assumed number of turns in primary, 
and / = frequency. 

The instantaneous value of the counter E.M.F. of self- 
induction will be (§ 13, vol. i., § 3) 

, _ _ T^d^' __TJ ($^ sin o)/) 
^ ~ 'dr~~ It ' 

/ = — Tp^^w cos 0)t, 

e^=— 2'nfT^,^,,, 
because the maximum value of the cosine is unity. 

V2 

At no load this is equal and opposite to the primary im- 





^, 


n= (^,n^, 




E = 


e 

108 ~ 


io-W2 7r/7; 


A(S>..- 




.-. A = 


lo'E 






I.41 7r/7;.(B„. 




. To obtain c 


and d \ 


'ji inches. 





136 ALTERXATIXG-CURREXT MACHINES. 

, A 

cd = , 

6-45 

c = n_LiL, 
2.54 

and d=-^- 

71 2.54 

III. To obtain the depth of coil icindiiig t^ and t^ in 
incites. 

Let d^ = diameter of priman^ wire, including insulation, 
in inches. 

d^ = diameter of secondar}- wire, including insula- 
tion, in inches, 

as found from a wire table ; then, allowing \ inch at each 
end for insulation, 

T 

— d: 

a 

2 

approximately, smce but half the primary is woimd on 

each hmb and 

tT; d: 
t. = — • 5 



2 I 
a 

2 

w^here r is the ratio of transformation, ~ 

The value of a is foimd in the next paragraph. 
IV. To obtain a and b in incJies. Evidentl}- the trans- 
former could not be assembled unless 

^ > 2 (/'^ + ^. + insulation and clearance). 
Assume /^ = 2 f 4, + /, 4- . 



Now 
and 

so 



THE TRANSFORMER. 

a = mb. 



t. = tL 



a = 2 7?Z 



4+T4(;7)+ir 



i] 



137 



But also, 



/„ = 



a 

2 



so substituting and transposing, 



a m\ \ a 



mTd} 



I +T 



All the terms of the right-hand member are known, so 
it may be reduced to a simple number, and set equal to K. 
Then 



\\ 2I 8 



and , = (7., + l)±y/(A'_|..) + (|..+iJ 



V. To obtain the volume v of iron in cubic centimeters. 
— About 90% of a volume occupied by laminated iron is 
metal. 

v=2{a-\rb-\-2c) X c X d X 2.54 X 0.9. 

VI. To obtain tJie watts P^, lost in hysteresis. — Accord- 
ing to Steinmetz's Law, using t] = .003, 

Hysteresis loss = .003 7^(B„/-^ ergs per cycle. 



T,. 



10^ 



138 ALTERNATING-CURRENT MACHINES. 

\'II. To obtaiJi the resistance of the secondary R^ i?i 
ohms. — Although surrounding a rectangular core, the coils 
are usually approximately circular in section, for con- 
venience in wmdinc^ and in insulatinsf. If the section of 
the core varies considerably from the square, allowance 
can be made in estimating the length of a mean turn. 

Considering the coil as truly cylindrical, and allowing |- 
inch insulation between it and the core, the length of a 
mean turn 






/ ^ I ^. 



The total length of primary wire (both limbs) is then tTJ, 
and its resistance can be found directly in a wire table 
giving the hot resistances of wires ; or, it mav be assumed 
that the transformer will operate at such a temperature 
that one mil foot has 1 1 ohms resistance, then 






41 8 2I 



12 X circular mils 

\lll. To obtain the resistance of the primary R^^ in 
ohms. — Similarlv to the above, the leno-th of a mean turn 



'O' 



C . I 



16 2/ 



allowins: vV inch insulation between the two coils, and the 

o 1 b 

total length of primary ^^-ire is TJ. 

The resistance can be found in a table, or calculated 
from 



i?„ = 



^ \i.4i 8 16 2/ 



^ 12 X circular mils 



THE TRANSFORMER. 139 

IX. To obtain tJie foucault current loss P^ in loatts. — 
Steinmetz has given the empirical formula 

where x is the thickness in mils of one lamina. Trans- 
former iron may be assumed to be from lo to 20 mils in 
thickness. 

X. To obtain tJie efficiency at any load, /^, in per cent. — 

EJ^ 

for a lighting transformer. If the load be inductive the 
term EJ^, whenever it occurs, must be multiplied by the 
power factor (cos <^). The error involved in the assump- 
tion I^= rl^ is negligible. 

After calculating the values in each of the preceding 
steps for the three values of 7^ and the three values of 
<^m suggested, the efficiency curve of each transformer 
should then be drawn, taking points at yL,i, i |, and full 
load. After having selected the most suitable, determine 
the following values. 

XL To determine the all-day efficiency in per cent. — 
The average lighting transformer is found to be loaded 
equivalent to full load for 5 hours, and no load for 19 
hours, per day. The all-day efficiency is 
7vatt hours output 



watt houi^s input 



per day. 



with non-inductive load, /«, being the full-load secondary 
current. 

XII. To determine the regulation in per cent. — In § 53 
was shown the method of calculating the magnetic leakage 



140 ALTERNATING-CURRENT MACHINES. 

of this type of transformer. Call the flux linking only the 
primary coils ^^ (this is twice that which hnks the coil of 
one limb of the transformer). Call that which links only 
the secondary coils $^. There is practically no voltage 
drop at no load, so E^= tE^. At full load there is a drop 
in the primary and in the secondary, due (a) to IR drop, 
(d) to self-induction caused by leakage flux. Knowing 
this leakage flux, by the formula of paragraph I., this sec- 
tion, calculate the voltage drop in primary and in secondary 
coils, thus, 

and ^,,= io-W2 7r/T7;,<l>, 

The regulation, expressed in per cent, is 
Regulation = ^. - EK^.. + ^j?,) + ^-^ + ^.^J .,,, 

where I^-=rI^, and is the full-load current. Regulation as 
stated refers to a non-inductive load. 



MOTORS. 141 



CHAPTER VII. 

MOTORS. 

60. Rotating Field. — Suppose an iron frame, as in Fig. 
Ill, to be provided with inwardly projecting poles, and that 
these be divided into three groups, arranged as in the dia- 
gram, poles of the same group 
being marked by the same 
letter. If the poles of each 
group be alternately wound 
in opposite directions, and be 
connected to a single source 
of E.M.F., then the resulting current 
would magnetize the interior faces al- 
ternately north and south. If the im- 
pressed E.M.F. were alternating, then 
the polarity of each pole would change ^ig- "^• 

with each half cycle. If the three groups of windings 
be connected respectively with the three terminals of a 
three-phase supply circuit, any three successive poles will 
assume successively a maximum polarity of the same 
sign, the interval required to pass from one pole to its 
neighbor being one-third of the duration of a half cycle. 
The maximum intensity of either polarity is therefore 
passed from one pole to the next, and the result is a rotat- 
ing field. If the frequency of the supply E.M.F. be/, and 
if there be / pairs of poles per phase, then the field will 




142 ALTERNATING-CURRENT MACHINES. 

make one complete revolution in - seconds. It will there- 

f V , ^ . 

fore make - = ^- complete revolutions per second. A 
p 6q ^ ^ 

rotating field can be obtained from any polyphase supply- 
circuit by making use of appropriate windings. 

6i. The Induction Motor If a suitably mounted hollow 

conducting cylinder be placed inside a rotating field, it will 
have currents induced in it, due to the relative motion be- 
tween it and the field whose flux cuts the surface of the 
cylinder. The currents in combination with the fmx will 
react, and produce a rotation of the cylinder. As the cur- 
rent is not restrained as to the direction of its path, all of 
the force exerted between it and the field will not be in 
a tangential direction so as to be useful in producing rota- 
tion. This difficulty can be overcome by slotting the 
cylinder in a direction parallel with the axis of revolution. 
Nor will the torque exerted be as great as it would be if 
the cylinder were mounted upon a laminated iron core. 
Such a core would furnish a path of low reluctance for the 
flux between poles of opposite sign. The flux for a given 
magnetomotive force would thereby be greater, and the 
torque would be increased. 

Induction motors operate according to these principles. 
The stationary part of an induction motor is called the statoi^ 
and the moving part is called the rotor. It is common 
practice to produce the rotating field by impressing E.M.F. 
upon the windings of the stator. There are, however, 
motors whose rotating fields are produced by the currents 
in the rotor windings. 

Fig. 1 1 2 shows the stator core and frame of a Westing- 
house induction motor, and Fig. 1 1 3 shows the same with 



MOTORS. 



143 



the windings in place. Each projection of the core does 
not necessarily mean a pole ; for it is customary to employ 
a distributed winding, there being several slots per pole 





Fig. 112. Fig. 113, 

per phase. Fig, 1 14 shows the rotor. The inductors are 
copper bars embedded in slots in the laminated steel core. 
The y are all connecte d, in parallel, to copper collars or 

short-circuiting rings, one at 
each end of the rotor. They 
offer but a very small resist- 
ance, and the currents induced 
in them are forced to flow in 
a direction parallel with the 
axis. The reaction against the 
field flux is therefore in a 
proper direction to be most 
rotation. A rotor or armature of 




Fig. 114. 

eflicient in producin 



this type is called a sqtnrrel cage. 

62. Principle of Operation of the Induction Motor. — If 

the speed of rotation of the field be V R. p. m. and that of 



144 ALTERNATING-CURRENT MACHINES. 

the rotor be V r. p. im., then the relative speed between a 
given inductor on the rotor and the rotating field will be 
V— V K. p. M. The ratio of this speed to that of the field, 

V- T' 
viz., — — = s, is termed the s/i'/y, and is generally ex- 
pressed as a per cent of the synchronous speed. If the 
flux from a single north pole of the stator be $ maxwells, 
then the effectived. J/. F. induced in a single rotor inductor 

jr 

is 2.22 p^ s — io~^ where/ represents the number of 
60 

pairs of revolving poles. The frequency of this induced 
E.M.F. is different from that of the E.M.F. impressed upon 
the stator. It is s times the latter frequency. The fre- 
quency would be zero if the rotor revolved in synchronism 
with the field, and would be that of the field current if the 
rotor were stationary. As the slip of modern machines is 
but a few per cent (2% to 15%), the frequency of the 
E.M.F. in the rotor inductors, under operative conditions, 
is quite low. The current which will flow in a given in- 
ductor of a squirrel-cage rotor is difficult to determine. All 
the inductors have E.M.F.' s in them, which at any instant 
are of different values, and in some of them the current 
may flow in opposition to the E.M.F. It can be seen, how- 
ever, that the rotor impedance is very small. As the im- 
pedance is dependent upon the frequency, it will be larger 
when the rotor is at rest than when revolving. It will re- 
duce to the simple resistance when the rotor is revolving in 
synchronism. Suppose a rotor to be running light without 
load. It will revolve but slightly slower than the revolving 
field, so that just enough E.M.F. is generated to produce 
such a current in the rotor inductors that the electrical 
power is equal to the losses due to friction, windage, and 



MOTORS. 



145 



the core and copper losses of the rotor. If now a me- 
chanical load be applied to the pulley of the rotor, the 
speed will drop, i.e., the slip will increase. The E.M.F. and 
current in the rotor will increase also, and the rotor will 
receive additional electrical power, equivalent to the increase 
in load. The induction motor operates in this respect 
like a shunt motor on a constant potential direct-current 
circuit. If the strength of the rotating field, which cuts 



350,000- 


sc^"" ^^ 






^----iL^;j^~^>^^/^ 


" "^^\ y^^ 


800,000- 




y\ \ 


250,000- 


^^iw-^"^ 


^•v y' \. \ 


IJ 

Zi 






/\x^ \ \ 





^^^^>?^ "^"^ / 


\. ^ V \ 


»-200,'>00- 



^^ ^^^->/ 


\\ V \ 


z 






3 


^^^^^^^^^x 


\ \ \* 1 


FOOT-PO 
1 




\ \ \\ 




1^^"*^^ 


'\. "^N \S \^ I 




""^„ 


^Sy,^ "s \^ \\ 1 


100,000- 




\>\ \\] 






""*- ^^V^^ V \*1 


50,000- 









90 BO 70 60 50 
1 1 1. 1 1 


40 30 20 lO^^ 



STAND STILL 



SLIP^ 

Fig. 115. 



SYNCHRONISM 



the rotor inductors, were maintained constant, the slip, 
the rotor E.M.F., and the rotor current would vary directly 
as the mechanical torque exerted. If the rotor resistance 
were increased, the same torque would require an increase 
of slip to produce the increased E.M.F. necessary to send 
the same current, but the strict proportionality would be 
maintained. The rotating magnetism, which cuts the rotor 
inductors, does not, however, remain constant under vary- 



146 ALTERNATING-CURRENT MACHINES. 

ing loads. As the slip increases, more and more of the 
stator flux passes between the stator and rotor windings, 
without linking them. This increase of magnetic leakage 
is due to the cross magnetizing action of the increased 
rotor currents. The decrease of linked field flux not only 
lessens the torque for the same rotor current, but also 
makes a greater slip necessary to produce the same cur- 
rent. The relation which exists between torque and slip 
for various rotor resistances is shown in Fig. 115, where 
the full lines represent torque, and the dotted lines current. 
An inspection of the curves shows that the maximum 
torque which a motor can give is the same for different 
rotor resistances. The speed of the rotor, however, when 
the motor is exerting this maximum torque, is different for 
different resistances. This fact is made use of in starting 
induction motors so that the starting current may not be 
excessive. Fig. 116 shows a General Electric Form L 




rotor. The winding is polar, and not of the squirrel-cage 
type. The impedance can therefore be easily calculated. 
The terminals of the windings are connected to a resistance 
carried on the rotor spider. When the rotor reaches a 



MOTORS. 147 

proper speed the resistance may be cut out by pushing a 
knob on the end of the shaft, as shown in diagram. This 
arrangement permits of a small starting current under 
load and a large torque. Squirrel-cage motors require 
several times full-load current to start under load. Fig. 




Fig. 117. 

1 1 7 shows a General Electric Co. form M rotor. The 
winding is the same as in the Form L, except that its ter- 
minals are brought out to three slip-rings. A starting 
resistance can be placed away from the motor and be con- 
nected with the rotor windings by means of brushes rubbing 
upon the slip-rings. 

63. The Transformer Method of Treatment. — It is cus-. 
tomary in theoretical discussions to consider the induction 
motor as a transformer. Evidently when the rotor is 
stationary the machine is nothing but a transformer, with 
a magnetic circuit so constructed as to have considerable 
magnetic leakage. When the rotor is moving, the machine 
still acts as a transformer ; but the ratio of transformation 
and the frequency of the E.M.F. in the rotor, are but .y 
times what they were with a stationary rotor, the mechani- 
cal load taking the place of the electric load on the secon- 



148 ALTERNATING-CURRENT MACHINES. 

dary of the transformer. Bearing these facts in mind, the 
motor may be treated exactly like the transformer. Con- 
sider one phase of a polyphase motor. The pressure im- 
pressed upon the stator is greater than the pressure which 
is operative in inducing E.AI.F. in the rotor. The differ- 
ence is due to the resistance, the hysteresis, the eddy 
currents, and the magnetic leakage of the stator. The 
pressure to overcome each should be subtracted from the 
impressed pressure in the proper phase relation to get the 
operative pressure. The equivalent inductance of the 
magnetic leakage can be calculated for different currents, 
as was the case in the transformer. The voltage induced 
in the rotor is st times the operative pressure of the stator 
where t is the ratio of transformation. The current which 
it produces is dependent in magnitude and phase upon the 
impedance of the rotor windings. From the power repre- 
sented by this current at the rotor pressure must be 
subtracted the power lost in resistance, eddy currents, hys- 
teresis, friction, and windage of the rotor. What remains 
is given out by the motor as useful mechanical power. It 
should not be forgotten that the frequency of the rotor 
currents is but s times that of the impressed voltage. 

64. Ratio of Transformation The ratio of transforma- 
tion in an induction motor is without appreciable effect 
upon its operation. For motors of the same capacity it is 
the practice of the General Electric Company to use the 
same squirrel-cage rotor for different voltages and different 
phases. The stator windings alone are altered. Forms L 
and M rotors are not changed for change of voltage, but 
must of course be altered for change of phase, as they are 
polar wound. A certain 4-pole, 3-phase, 60-cycle, no- 



MOTORS. 



149 



volt, i-horse-power, General Electric induction motor has 
36 slots in the stator, each slot containing 20 conductors of 
size No. 13. The rotor contains 37 slots, each one con- 
taining one No. 2 wire. The slots are staggered by an 
amount equal to the distance between centers of two con- 
secutive slots. The rotor inductors are connected to short- 
circuit disks, one on each end of the rotor. 

65. Behavior of Induction Motors The relations be- 
tween speed, torque, power factor, efficiency, and current 
in the case of a typical induction motor operating under 
normal conditions is represented in Fig. 1 1 8. 

If the voltage impressed upon an induction motor be 
increased, there will result a proportional increase in the 
flux linked with the rotor, and in consequence a propor- 



^ 








?0F 


sync'hron 


SM 


— 






— . 












^ 


<^ 


Hi 




. — 






-..^ 


5^ 


^ 








"~~~~ 






/ 


/ 


^ 


^:. 


0^ 








■^^ 




) 












/ 


/ 




^^ 


VOV*- 












\ 


s. 




/ 


/ 


/ 


f 


















) 




/ 


/} 


/ 
























// 


/ 


























// 


f — 




75 F 


lORSE 


POWE 


^ MOTOR 








1/ 






( 


50 C 


YCL 


.ES 


220 


VO 


-TS 










\l 



























1-0 2.0 30 40 50 60 70 80 9.0 100 11.0 120 H. P. 

Fig. ii8. 

tional increase in the rotor current. As the torque de- 
pends upon the product of the flux and the rotor ampere 
turns, it follows that the^orque varies as the square of the 
impressed voltage. The capacity of a motor is therefore 



ISO ALTERNATING-CURRENT MACHINES. 

changed when it is operated on circuits of different volt- 
ages. 

Owing to the low-power factor of induction motors, 
transformers intended tOv supply current for their operation 
should have a higher rated capacity than that of the mo- 
tors. It is customary to have the kilowatt capacity of the 
trai^sforraer equal to the horse-power capacity of the motor. 

The low power-factor is due to magnetic leakage, i.e., 
flux linked with the stator, but not with the rotor wind- 
ings. This leakage increases with increase of length of 
air gap. It is hence desirable to have the gap as small as 
consistent with mechanical clearance. Concentricity of 
rotor and stator is to be obtained by making the bearings 
in the form of end plates fastened to the stator frame. 
Some makers send wedge gap-gauges with their machines 
so that a customer may test for eccentricity due to wear 
of the bearings. A small air gap, besides lowering the 
leakage and raising the power factor, increases the effi- 
ciency and capacity of the motor. 

The torque exerted on a constant loaded rotor is con- 
tinuous and constant in the case of a polyphase motor. 

The Stanley Company raise the power factor of their 
two-phase motors to nearly unity by using condensers to 
neutralize the lag produced by leakage. 

The direction of rotation of a three-phase motor can be 
changed by transposing the supply connections to any two 
terminals of the motor. In the case of a two-phase, four- 
wire motor, the connections to either one of the phases, 
may be transposed. 

66. Starting of Squirrel-Cage Motors To avoid the 

excessive rush of current which would result from connec- 



MOTORS. 



151 



tion of a loaded squirrel-cage motor to a supply circuit, use 
is made by both the Westinghouse Company and the 
General Electric Company of starting compensators. These 
are autotransformers which are connected between the 
supply mains, and which, through taps, furnish to the 
motor circuits currents at a lower voltage than that of 
the supply mains. After the rotor has attained the speed 
appropriate to the higher voltage, the motor connections are 
transferred to the mains, and the compensator is thrown 



Generator 




1 



Running Side 



CH OJ 



CM O-l O-J O-l 



Starting Side 




Q 



M 



? 



CompensatoT-windinA 

Fig. 119. 

out of circuit. The connections are shown in Fig. 119, 
and the appearance of the General Electric Company com- 
pensator is shown in Fig. 120. The change of connec- 
tions is accompUshed by moving the handle shown at the 
right of the figure. While the compensator is supphed 
with various taps, only that one which is most suitable for 
the work is used when once installed. The Westinghouse 
compensator is shown in Fig. 121. When the handle is 
down on one side, the autotransformers are in circuit, and 



152 ALTERXATIXG-CURREXT MACHINES. 

the motor is connected with the low-voltage taps. Upon 
throwing over the switch the transformers are cut out, and 
the motor is connected directly with the mains. 




Fig. I20. 

Where special step-down transformers are used for indi- 
vidual motors, or where several motors are located close to 
and operated from a bank of transformers, it is sometimes 
practical to bring out taps from the secondary \\inding, and 
use a double-throw motor switch, thereby making pro\ision 
for starting the motor at low voltage, while avoiding the 
necessity for a starting compensator. 

The General Electric Company make small squirrel- 
cage motors, with centrifugal friction clutch pulleys ; so 
that although a load may be belted to the motor, it is not 
applied to the rotor until the latter has reached a certain 
speed. The starting current is therefore a no-load starting 
current. 



MOTORS. 



153 




Fig. 121. 



67. Phase Splitters In order to operate polyphase in- 
duction motors upon single-phase circuits, use is made of 
inductances in series with one motor circuit to produce a 



154 



ALTERNATING-CURRENT MACHINES. 




Fig. 122. 

lagging current, or of condensers to produce a leading cur- 
rent, or of both — one in each of two legs.. The General 
Electric Company, in its condenser compensator, for use 
with small motors, as shown in Fig. 122, employs an 
autotransformer and condenser connected, 
as in diagram Fig. 123. 

The autotransformer is used to step-up 
the voltage, which is impressed upon the 
condenser, to 500 volts. The necessary 
size of the condenser is thereby reduced. 
The equivalent impedance of the auto- 
transformer and condenser, as connected, 
is such as to produce a leading current in 
the one-phase sufficient to give a satisfac- 
tory starting torque, and it brings the power 
factor practically up to unity at all loads. p^g^ ^^^^ 




68. Single-Phase Induction Motors. — A two -phase induc- 
tion motor will operate fairly well, if, af Fer attaining full 



MOTORS. 



155 



speed, one of the two phases be disconnected from the sup- 
ply circuit. it_will not start from rest under the influence 
of the one-p hase excitation. The load remaining constant, 
the one phase will take twice its original current. Simi- 
larly, a three-jDhase motor will operate well upon one-phase 
excitation. The current in this case will be 1.5 times 
what it previously was. A motor consisting of a rotor- 




Fig. 124. 

and a stator-wound single phase will, in a like manner, 
operate satisfactorily when once started. In the Wagner 
single-phase induction motor (Fig. 124), the rotor windings 
are connected to a commutator, with its brushes joined 
together by a conductor of low resistance. The stator 
is supplied with single-phase excitation. The rotor is 
brought up to speed by the reaction between the current 
which is induced in the rotor windings and the. stator flux. 



156 ALTERXATIXG-CURREXT MACHINES. 

Upon reaching speed, a centrifugal de\'ice, sho^Ti in the 
figure, causes the commutator bars to be short-circuited, 
and the brushes are simultaneously lifted from the commu- 



.0 10 23 30 40 50 60 70 sa so 100 

j^FULLUMD 

Fig. 125. 

tator. Tests have been made upon this type of motor at 
various universities, including Harvard, University of Illi- 
nois, and Purdue Universit}'. The results are concordant, 

and are represented in the curves Fig. 125. 

69. The Monocyclic System. — This is a system advocated 
by the General Electric Company for the use of plants 
whose load is chiefly lights, but which contains some 
motors. The monocyclic generator is a modified single- 
phase alternator. In addition to its regular winding, it 
has a so-called teazer ^\-indLng, made of wire of suitable 
cross section to carrv the motor load, and with enousrh 
turns to produce a voltage one-fourth that of the regular 
^^'inding, and lagging 90° in phase behind it. One end of 
the teazer winding is connected to the middle of the r^;ii- 



MOTORS. 157 

lar winding, and the other end is connected through a sHp- 
ring to a third hne wire. 

A three-terminal induction motor is used, the terminals 
being connected to the line wires either directly or through 
transformers. 

70. Frequency Changers These are machines which 

are used to transform alternating currents of one frequency 
into those of another frequency. They are commonly used 
to transform from a low frequency (say from 25 or 40) to 
a higher one. They depend for their operation upon the 
variation with slip of the frequency of the rotor E.M.F.'s 
of an induction motor. The common practice for raising 
the frequency is, to have a synchronous motor turn the 
rotor of an induction motor in a direction opposite to the 
direction of rotation of the latter's field. The synchronous 
motor and the stator windings of the induction motor are 
connected to the low frequency supply mains. Slip-rings 
connected to the rotor windings of the induction motor 
supply current at the higher frequency. The size of the 
synchronous motor necessary to drive the frequency 
changer is the same percentage of the total output as the 
rise of frequency is to the higher frequency. 

71. Speed Regulation of Induction Motors. — The speed 
of an induction motor can be varied by altering the voltage 
impressed upon the stator, by altering the resistance of the 
rotor circuit, or by commutating the stator windings so as 
to alter the multipolarity. The first two methods depend 
for their operation upon the fact that, inasmuch as the 
motor torque is proportional to the product of the stator 
flux and the rotor current, for a given torque the product 



158 ALTERNATING-CURRENT MACHINES. 

must be constant. Lessening the voltage impressed upon 
the stator lessens the flux, and also the rotor current, if the 
same speed be maintained. The speed, therefore, drops 
until enough E.M.F. is developed to send sufficient current 
to produce, in combination with the reduced flux, the 
equivalent torque. Increasing the resistance of the rotor 
circuit decreases the rotor current, and requires a drop in 
speed to restore its value. Both of these methods result 
in inefficient operation. If the impressed voltage be re- 
duced, the capacity of the motor is reduced. In fact, the 
capacity varies as the square of the impressed voltage. 
Changes in the multipolarity of the stator requires compli- 
cated commutating devices. 

72. Synchronous Motors. — Any excited single-phase 
or polyphase alternator, if brought up to speed, and if con- 
nected with a source of alternating E.M.F. of the same 
frequency and approximately the same pressure, will ope- 
rate as a motor. The speed of the rotor in revolutions 
per second will be the quotient of the frequency by the 
number of pairs of poles. This is called the synchronous 
speed ; and the rotor, when it has this speed, is said to be 
running in synchronism. This exact speed will be main- 
tained throughout wide ranges of load upon the motor up 
to several times full-load capacity. 

To understand the action of the synchronous motor, 
suppose it to be supplied with current from a single 
generator. 

Let E^ = E.M.F. of the generator, 

E.^ = E.M.F. of the motor at the time of connec- 
tion with the generator, 
B — Phase angle between E^ and E.-,, 



MOTORS. 159 

R = Resistance of generator armature, plus that 
of the connecting wires and of the 
motor armature, and 
(joZ = Reactance of the above. 

The resultant .£".y]/.i^., E which is operative in sending 
current through the complete circuit, is found by combin- 
insf E, and E^ with each 
other at a phase differ- 
ence 9, as in Fig. 126. 

Representing the 
angle between E^ and E 
and E,^ and E hy a and Fig, 126. 

/3 respectively, it follows that 

E = E^ cos a + E^ cos fS. 

This resulting E.M.F. sends through the circuit a 
current whose value is 




and it lags behind E by an angle <^, such that tan </> = —-. 

R 

The power P^ which the generator gives to the circuit is 

P^ = EyI cos (a — ^) 

and the power P.^ which the motor gives to the circuit is 

P,^ = EUcos (/? + <^). 

Now, if in either of the above expressions for power, the 
cosine has any other value than unity, then the power 
will consist of energy pulsations, there being four pulsa- 
tions per cycle. The energy is alternately given to and 
received from the circuit by the machine. If the cosine 
be positive, the amount of energy in one pulsation, which 



l6o ALTERNATING-CURRENT MACHINES. 

is given to the circuit, will exceed the amount in one 
of the received pulsations. The machine is then acting 
as a generator. If the cosine be negative the opposite 
takes place, and the machine operates as a motor. As a 
and /? are but functions of E^, E^, and 6, and as these latter 
are the quantities to be considered in operation, it is desir- 
able to eliminate the former. By a somewhat complicated 
analytical transformation it can be shown that 

P, = , ^'^' cos (^ + c^) + , 3' cos «/,, 



and P^ = ,- ' ' cos (^ - c^) + - ' cos (^. 



££^^ cos (^ - c^) + ^^^ 

If there were no losses due to resistance, etc., P^ would be 
numerically exactly equal to P^. Neglecting any losses 
in the machines, except that due to resistance, the alge- 
braic sum of P^ and P^ is equal to R/^. In order to 
determine the behavior of a synchronous motor when on 
a given circuit, use is made of the above formula for power, 
and each case must be considered by itself. The method 
of procedure is shown in the next article. 

73. Special Case. — Suppose a single-phase synchronous 
motor, excited so as to generate 2100 volts, to be con- 
nected to a generator giving 2200 volts, the total resis- 
tance of the circuit being 2 ohms and the reactance i ohm. 
Then the angle <^ of current lag behind the resultant 

E.MP\ has a value tan <^ = — — = 0.5, whence <^ = 26° 34'. 

R 

A preliminary calculation, using the formulas of the pre- 
vious article, shows that both machines act as generators 
for values of ^between 0° and 120°, and between 240° and 
360'' approximately. 



MOTORS. 



i6i 



Calculations of P^ and P^ for various values of B between 
120° and 240° have been made, and are embodied in the 
form of curves in Fig. 127. From an inspection of these 




Fig. 127. 

curves, and a consideration of the equations from which 
the curves are derived, the following conclusions may be 
drawn : — 

{a) The motor will operate as such for values of be- 
tween 175° and 238°. The difference between these 
angles may be termed the operative range. 

{b) The generator would operate as a motor for values 
of between 133° and 174°, providing the motor were 
mechanically driven so as to supply the current and power ; 
i.e., what was previously the motor must now operate as 
a generator. 

{c) The motor, within its operative range, can absorb 
any amount of power between zero and a certain maxi- 
mum. To vary the amount of received power, the motor 
has to but slightly shift the phase of its E.M.F. in respect 
to the impressed E.M.F., and then to resume running in 



l62 ALTERXATIXG-CURREXT MACHINES. 

synchronism. The sudden shift of phase under change 
of load is the fundamental means of power adjustment in 
the synchronous motor. It corresponds to change of slip 
in the induction motor, to change of speed in the shunt 
motor, and to change of magnetomotive force in the 
transformer. 

(d) For all values of the received power, except the 
maximum, there are two values of phase difference 6. 
At one of these phase differences more current is required 
for the same power than at the other. The value of the 
current in either case can be calculated as follows : — 
Since P^^ F.^ = RJ'^- 



The values of /are plotted in the diagram. The efficiency 

P 

of transmission ^ = -^ is also different for the two values 

1 
of ^ . It is also represented by a curve. 

If the phase alteration, produced by an added mechan- 
ical load on the motor, results in an increase of power 
received by the motor, the running is said to be stable. If 
on the other hand, the increase of load produces a decrease 
of absorbed power, the running is unstable. 

{e) If for any reason the phase difference B, between 
the E.M.F.'s of the motor and generator, be changed to a 
value without the operative range for the motor, the motor 
will cease to receive as much energy from the circuit as it 
gives back, and it will, therefore, fall out of step. Among 
the causes which may produce this result are sudden 
variations in the frequency of the generator, variations in 
the angular velocity of the generator, or excessive me- 



MOTORS. 163 

chanical load applied to the motor. In slowing down, all 
possible values of will be successively assumed ; and it 
may happen that the motor armature may receive suffi- 
cient energy at some value of to check its fall in speed, 
and restore it to synchronism, or it may come to a stand- 
still. 

(/) Under varying loads the inertia of the motor 
armature plays an important part. The shifting from one 
value of to another, which corresponds to a new mechan- 
ical load, does not take place instantly. The new value is 
overreached, and there is an oscillation on both sides of 
its mean value. This oscillation about the synchronous 
speed is termed hiniting. If the armature required no 
energy to accelerate or retard it, this would not take 
place. 

(^g) The maximum negative value of P^ — that is, the 
maximum load that the motor can carry — is evidently when 
cos ( ^ — (^ ) = — I or when ^ — <^ = 180°. The formula 
for the power absorbed by the motor then reduces to 

„ E^ cos <i — E.E^ 
Pirn = , ^^^ = 320 K. w. 

(Ji) The operative range of the motor can be determined 
by making P,^ equal to zero. By transformation the for- 
mula then becomes 

//I ,N E 2^ COS cf> 

cos (^ - <^) = - -~^ • 

Two values of (0 — (/>) result, one on each side of 180°. 
In the case under consideration cos (^ — <^) = — .851 and 
- ct>= 211° 40' or 148° 20^ Since c^ - 26° 34', - 238° 
14' or 174° 54^ 



1 64 



ALTERNATING-CURRENT MACHINES. 



74. The Motor E.M.F. — To determine what value of 
E^ will give the maximum value of power to be absorbed 
by a motor, consider E.^ as a variable in the equation given 
in {g) above. 

Differentiating 

dE.-. 



= 2 E.2 cos (^ — E^, 



and setting this equal to zero and solving, 

^1 



E.= 



volts. 




CURRENT LAGGING E, 

Fig. 128. 



2 COS (^ 

At this voltage the maximum possible intake of the motor 
is 6 1 1 K. w. If the voltage of the motor be above this or 
below it, its maximum intake will be smaller. 

Remembering that the current lags behind the resultant 
pressure of the generator and motor pressures by an angle 
<^, which is solely dependent upon w, 
Z, and R, it will be easily seen, from 
an inspection of Figs. 128, 129, and 
1 30, that the current may be made 
to lag behind, lead, or be in phase 
^^ith E^, by simply altering the value 
of E.,. This may be done by vary- 
ing the motor's field excitation. A 
proper excitation can produce a unit 
power factor in the transmitting 
line. The over-excited synchronous 
motor, therefore, acts like a con- 
denser in producing a leading cur- 
rent, and can be made to neutralize the effect of induct- 
ance. The current which is consumed by the motor for a 
given load accordingly varies with the excitation. The 




CURRENT LEADING E, 

Fig. 129. 



CURRENT IN PHASE WITH E, 

Fig. 130. 



MOTORS. 



165 



relations between motor voltage and absorbed current for 
various loads are shown in Fig. 131. 

Synchronous motors are sometimes used for the purpose 
of regulating the phase relations of transmission lines. 




Fig. 131. 

The excitation is varied to suit the conditions, and the 
motor is run without load. Under such circumstances the 
machines are termed synchronoiLS compensators. 

The capacity of a synchronous motor is limited by its 
heating. If it is made to take a leading current in order 
to adjust the phase of a line current, it cannot carry its 
full motor load in addition without heating. 

75. Polyphase Synchronous Motor. — The discussion 
which has just been given applies to the single-phase 
motor. The facts brought out are equally applicable to the 
polyphase motor. In the latter case each leg or phase is 
to be considered as a single-phase circuit. The total power 
is that of each phase multiplied by the number of phases. 

76. Starting Synchronous Motors. — These motors do not 
have sufficient torque at starting to satisfactorily come up 



1 66 



ALTERXATIXG-CURREXT MACHINE; 



to speed under load. They are, therefore, preferably 
brought up to synchronous speed by some auxiliary source 
of power. In the case of pohphase systems an induction 
motor is very satisfactory. Its capacity need be but ^^ that 
of the large motor. Fig. 132 shows a 750 k. w. quarter- 
phase General Electric motor with a small mduction motor 



% Ik. 




^I 



Fig. 132. 

geared to the shaft for this purpose. This motor may be 
mechanically disconnected after synchronism is reached. 
Before connection of the synchronous motor to the mains 
it is necessar}- that the motor should not only be in s}!!- 
chronism, but should have its electromotive force at a 
difference of phase of about 180^ with the impressed 
pressure. To determine both these points a simple device, 



MOTORS. 167 

known as a synchroniser, is employed. It consists of 
an incandescent lamp connected in series with the sec- 
ondaries of two transformers, whose primaries are con- 
nected respectively with the line and with the motor 
terminals. The brightness with which the lamp glows 
is a measure of the phase difference between the two 
E.M.F.'s. It is customary to so connect the transformers 
that when the motor E.M.F. is at 180° with the line 
pressure, the lamp will have its greatest brilliancy. As 
the motor is coming up to speed, the lamp will be 
alternately bright and dark. The alternations will grow 
slower as synchronism is approached, and will finally be 
so slow as to permit the closing of the main switch at the 
proper instant. 

Synchronous motors may be brought up to speed with- 
out any auxiliary source of power. The field circuits are 
left open ; and the armature is connected either to the full 
pressure of the supply, or to this pressure reduced by 
means of a starting compensator, such as was described 
in § 66. The magnetizing effect of the armature ampere 
turns sets up a flux in the poles sufficient to supply a small 
starting torque. When, after running a sufficient time as an 
induction motor, synchronism is nearly attained, the fields 
may be excited and the motor will come into step. The 
load is afterwards applied to the motor through friction 
clutches or other devices. There is great danger of per- 
forating the insulation of the field coils when starting in 
this manner. This is because of the high voltage produced 
in them by the varying flux. In such cases each field 
spool is customarily open-circuited on starting. Switches 
which are designed to accomplish this purpose are called 
break-up switches. 



l68 ALTERNATING-CURRENT MACHINES. 

77. Parallel Running of Alternators Any two alter- 
nators adjusted to have the same E.M.F., and the same 
frequency, may be synchronized and run in parallel. Ma- 
chines of low armature reaction have large synchronizing 
power, but may give rise to heavy cross currents, if thrown 
out of step by accident. The contrary is true of machines 
having large armature reaction. Cross currents due to 
differences of wave-form are also reduced by large arma- 
ture reaction. The electrical load is distributed between 
the two machines according to the power which is being 
furnished by the prime movers. This is accomplished, as 
in the case of the synchronous motor, by a slight shift of 
phase between the E.M.F.'s of the two machines. The 
difficulties which have been experienced in the parallel 
running of alternators have almost invariably been due to 
bad regulation of the speed of the prime mover. Trouble 
may arise from the electrical side, if the alternators are 
designed with a large number of poles. Composite wound 
alternators should have their seiies compounding coils con- 
nected to equalizing bus bars, the same as compound wound 
direct-current generators. 



CONVERTERS. 



169 



CHAPTER VIII. 



CONVERTERS. 



78. The Converter. — The converter is a machine hav- 
ing one field, and one armature, the latter being supplied 
with both a direct-current commutator and alternating- 
current slip-rings. When brushes, which rub upon the 
slip-rings, are connected with a source of alternating 
current of proper voltage, the armature will rotate syn- 
chronously, acting the 
same as the armature of a 
synchronous motor. While 
so revolving, direct current 
can be taken from brushes 
rubbing upon the commu- 
tator. The intake of cur- 
rent from the alternating- 
current mains is sufficient 
to supply the direct-current 
circuit, and to overcome 
the losses due to resistance, 
friction, windage, hyster- ^^^' ^^^' 

esis, and eddy currents. The windings of a converter 
armature are closed, and simply those of a direct-current 
dynamo armature with properly located taps leading to the 
slip-rings. Each ring must be connected to the armature 
winding by as many taps as there are pairs of poles in 
the field. These taps are equidistant from each other. 




I/O ALTERXATIXG-CURREXT .AIACHIXES. 

There may be any number of rings greater than one. 
A converter having n rings is called an /^-ring converter. 

The taps to successive rings are -th of the distance be- 

71 

tween the centers of two successive north poles from each 
other. Fig. 133 shows the points of tapping for a 3-ring 
multipolar converter. 

A converter may also be supplied with direct current 




■Fig. 134. 

through its commutator, while alternating current is taken 
from the slip-rings. Under these circumstances the 
machine is termed an inverted eonverter. Converters are 
much used in lighting and in power plants, sometimes 
receiving alternating current, and at other times direct 
current. In large city distributing systems they are often 
used in connection with storage batteries to charge them 



CONVERTERS. I/I 

from alternating-current mains during periods of light 
load, and to give back the energy during the heavy load. 
They are also used in transforming alternating into direct 
currents for electrolytic purposes. A three-phase machine 
for this purpose is shown in Fig. 1 34. 

A converter is sometimes called a rotary converter or 
simply a rotary. 

79. E.M.F. Relations. — In order to determine the re- 
lations which exist between the pressures available at the 
various brushes of a converter, 

Let E^i = the voltage between successive direct-current 
brushes. 
£,^ = the effective voltage between successive rings 
of an n-ring converter. 

a = the maximum E.M.F. in volts generated in a 
single armature inductor. This will exist 
when the conductor is under the center of a 
pole. 

b =■ the number of armature inductors in a unit 
electrical angle of the periphery. The 
electrical angle subtended by the centers of 
two successive poles of the same polarity 
is considered as 2 7r 

The E.M.F. generated in a conductor may be considered 
as varying as the cosine of the angle of its position relative 
to a point directly under the center of any north pole, the 
angles being measured in electrical degrees. At an angle 
y8, Fig. 135, the E.M.F. generated in a single inductor G 
is a cos ^ volts. In an element d^ of the periphery of 
the armature there are bd^ inductors, each with this 
E.M.F. If connected in series they will yield an E.M.F. 



1/2 



ALTERNATING-CURRENT MACHINES. 



of ab cos /? d^ volts. The value of ab can be determined 
if an expression for the E.M.F. between two successive 

direct-current brushes be 
V^S^^ determined by integration, 
and be set equal to this 
value Ea as follows : 







ab cos ^d^ = 2 ab. 



2 



i-lg. 135. 



27r 



In an ;/-ring converter, the 
electrical angular distance 
between the taps for two 

The maximum E.M.F. will be 



generated in the coils between the two taps for the succes- 
sive rings, when the taps are at an. equal angular distance 
from the center of a pole, one on each side of it, as shown 
in the figure. This maximum E.M.F. is 



V2 ^„ = 



/■ 



ah CCS ^d^ = 2 ab sin 



E,i sin 



The effective voltage between the successive rings is 
therefore 

En = — i^ Sin - . 

By substituting numerical values in this formula, it is 
found that the coefficient by which the voltage between 



CONVERTERS. 



173 



the direct -current brushes must be multipUed in order to 
get the effective voltage between successive rings is for 

2 rings ,0.707 

3 rings 0.612 

4 rings 0.500 

6 rings 0.354 

In practice there is a slight variation from these co-effi- 
cients due to the fact that the air-gap flux is not sinusoid- 
ally distributed. 



80. Current Relations. — In the following discussion it is 
assumed that a converter has its field excited so as to 
cause the alternating currents in the armature inductors to 
lag 180° behind the alternating E.M.F. generated in them. 

The armature coils carry currents which vary cyclically 
with the same frequency as that of the alternating-current 
supply. They differ 
widely in wave-form from 
sine curves. This is be- 
cause they consist of two 
currents superposed upon 
each other. Consider a 
coil B, Fig. 136. It car- 
ries a direct current whose 

value — is half that car- 

2 

ried by one direct-current 
brush, and it reverses its 
direction every time that 
the coil passes under a 




Fig. 136. 

brush. The coil, as well as 
all others between two taps for successive slip-rings, also 

This current has its zero 



carries an alternating current. 



174 



ALTERXATINXx-CURREXT MACHINES. 



value when the pomt A, which is midway between the 
successive taps, passes under the brush. The coil being 
\J/ electrical degrees ahead of the point A, the alternating- 
current will pass throuo-h zero — of a cvcle later than 

the direct current. The time relations of the two currents 
are shown in Fig. 137. 

To determine the maximum value of the alternating 
current consider that, after subtracting the machine losses. 




Fig. 137- 

the alternating-current power intake is equal to the direct- 
current power output. Neglecting these losses for the 
present, if E., represents the pressure and I„ the effective 
alternating current in the armature coils between the suc- 
cessive slip-rings, then for the parts of the armature wind- 
ings covered by each pair of poles 

E, . TT ^ 

Therefore, the maximum value of the alternating current is 



V27;, = 



/. 



;/ sin 



The time variation of current in the particular coil B is 
obtained by taking the algebraic sum of the ordinates of 



CONVERTERS. 



175 



the two curves. This yields the curve shown in Fig. 138. 
Each inductor has its own wave-shape of current, depend- 
ing upon its angular distance i// from the point A. 




Fig, 138. 

81. Heating of the Armature Coils The heating ef- 
fect in an armature coil due to a current of such peculiar 
wave-shape as that shown in Fig. 138 can be determined 
either graphically or analytically. The graphic determina- 
tion requires that a new curve be plotted, whose ordinates 
shall be equal to the squares of the corresponding current 
values. The area contained between this new curve and 
the time axis is then determined by means of a planimeter. 
The area of one lobe is proportional to the heating value 
of the current. This value may be determined for each 
of the coils between two successive taps. An average of 
these values will give the average heating effect of the 
currents in all the armature coils. The heating is different 
in the different coils. It is a maximum for coils at the 
points of tap to the slip-rings and is a minimum for coils 
midway between the taps. 

The analytical determination is made as follows : For 
a coil which is i// electrical degrees from a point midway 



176 ALTERNATING-CURRENT MACHINES. 

between successive slip-ring taps, at the time t seconds 
after passing a direct-current positive brush, the instanta- 
neous value of the current is 



r =^ r 4 sin (2 Trft - xf) _ ^ ] 



The effective value of the current in this coil is, therefore, 



^ ^ / _ 16 cos ^ 



ir?i sm - ;r sm'' - 
n n 

where Q represents for simplicity the value of the radical. 
The heating, due to the current in this coil, is propor- 

/2^2 

tional to '^ ^ , and the average heating over the whole 

4 
armature is proportional to 



l7rj_n 4 



<?^^ = ^" 




Inasmuch as the heatmg of this armature when run as a 

/ ^ 
simple direct-current generator is proportional to -^, it 

4 
can, with the same heating, when operating as a conver- 
ter, put out — - as much direct current. 



V^ 



2 . ., TT 

71 sm - 



CONVERTERS. 177 

82. Capacity of a Converter By inserting numerical 

values in the above equation it is found that a machine has 
different capacities, based upon the same temperature rise, 
according to the number of slip-rings, as shown in the fol- 
lowing table. The armature is supposed to have a closed- 
coil winding : — 

CONVERTER CAPACITIES. 
Used as a Kilowatt Capacity 

Direct-current generator 100 

Single-phase converter 85 

Three-phase converter 134 

Four-phase converter 164 

Six-phase converter 196 

Twelve-phase converter 227 

The overload capacity of a converter is limited by com- 
mutator performance and not by heating. As there is but 
small armature reaction, the limit is much higher than is 
the case with a direct-current generator. 

83. Starting a Converter. — Converters may be started 
and be brought up to synchronism by the same methods 
which are employed in the case of synchronous motors. 
It is preferable, however, that they be started from the 
direct-current side by the use of storage batteries or other 
sources of direct current. They may be brought to a little 
above synchronous speed by means of a starting resistance 
as in the case of a direct-current shunt motor, and then, 
after disconnecting and after opening the field circuit, the 
connections with the alternating-current mains may be 
made. This will bring it into step. 

84. Armature Reaction. — The converter armature cur- 
rents give rise to reactions which consist of direct-current 



178 



ALTERNATING-CURRENT MACHINES. 



generator armature reactions superposed upon synchronous 
motor armature reactions. It proves best in practice to 
set the direct-current brushes so as to commutate the cur- 
rent in coils when they are midway between two succes- 




Fig. 139- 

sive poles. The direct-current armature reaction, then, con- 
sists in a cross-magnetization which tends to twist the field 
flux in the direction of rotation. When the alternating 
currents are in phase with the impressed E.M.F. they also 
exert a cross-magnetizing effect which tends to twist the 



CONVERTERS. 179 

field flux in the opposite direction. The result of this neu- 
tralization is a fairly constant distribution of flux at all 
loads. Within limits even an unbalanced polyphase con- 
verter operates satisfactorily. There is no change of field 
excitation necessary with changes of load. 

The converter is subject to hunting the same as the 
synchronous motor. As its speed oscillates above and 
below synchronism, the phase of the armature current, in 
reference to the impressed E.M.F., changes. This results 
in a distortion of the field flux, of varying magnitude. 
This hunting is much reduced by placing heavy copper 
circuits near the pole horns so as to be cut by the oscillat- 
ing flux from the two horns of the pole. The shifting of 
flux induces heavy currents in these circuits which oppose 
the shifting. Fig. 139 shows copper bridges placed be- 
tween the poles of a converter for this purpose. 

When running as an inverted converter from a direct- 
current circuit, anything which tends to cause a lag of the 
alternating current behind its E.M.F. is to be avoided. 
The demagnetization of the field by the lagging current 
causes the armature to race the same as in the case of an 
unloaded shunt motor with weakened fields. Converters 
have been raced to destruction because of the enormous 
lagging currents due to a short circuit on the alternating- 
current system. 

85. Regulation of Converters The field current of a 

converter is generally taken from the direct-current 
brushes. By varying this current the power factor of the 
alternating-current system may be changed. This may 
vary, through a limited range, the voltage impressed 
between the slip-rings. As the direct-current voltage 



i8o 



ALTERNATING-CURRENT MACHINES. 



Regulator; 



bears to the latter a constant ratio it may also be varied. 

This is, however, an uneconomical method of regulation. 

Converters are usually fed through step-down transform- 
ers. In such cases 
there are two com- 
mon methods of regu- 
lation, which vary the 
voltage supplied to 
the converter's slip- 
of Stillwell, which is 




Fig. 140. 

rings. The first is the method 
shown in the diagram. Fig. 140. 



The regulator consists of a transformer with a sectional 



.,^-#-N, 




Fig. 141. 



CONVERTERS. l8l 

secondary. Its ratio of transformation can be altered by 
moving a contact-arm over blocks connected with the 
various sections, as shown in the diagram. The primary 
of the regulator is connected with the secondary terminals 
of the step-down transformer. The sections of the second- 
ary, which are in use, are connected in series with the step- 
down secondary and the converter windings. 

The second method of regulation is that employed by 
the General Electric Co. The ratio of transformation of 
a regulating transformer, which is connected in circuit in 
the same manner as the Stillwell regulator, is altered by 
shifting the axes of the primary and secondary coils in 
respect to each other. Fig. 141 shows such a transformer, 
the shifting being accomplished by means of a small, 
direct-current motor mounted upon the regulator. The 
primary windings are placed in slots on the interior of a 
laminated iron frame, which has the appearance of the 
stator of an induction motor. The secondary windings are 
placed in what corresponds to the slots of the rotor core. 
The winding is polar ; and if the secondary core be rotated 
by an angle corresponding to the distance between two 
successive poles, the action of the regulator will change 
from that of booster to that of crusher. 



1 82 ALTERNATING-CURRENT MACHINES. 



CHAPTER IX. 

POWER TRANSMISSION. 

86. Superiority of Alternating Currents. — The two 

great sources of energy for use in manufacturing establish- 
ments and in land transportation systems are the coal 
mines and the water powers. While coal can be trans- 
ported to the point of utilization of the energy, the 
energy of the waterfall cannot be commercially transmitted 
to a. long distance without the use of electricity. In 
many cases it is uncertain whether it is not cheaper to 
transmit the energy of the coal in the form of electrical 
energy than to transport the coal itself. There is gen- 
erally greater convenience and greater flexibility in the 
application and utilization of the transmitted electrical 
energ}\ 

The electrical transmission can be accomplished by 
means of direct currents or by means of alternating cur- 
rents. F'or transmission over anything but quite short 
distances the alternating current is preferable to the direct 
current. Even for short distances, when these pass 
through densely populated districts, the alternating cur- 
rent is adopted for pure transmission purposes. 

The direct current has its points of superiority. Its 
use is not attended by inductive disturbances with the ac- 
companying drop and sometimes low power factor ; it is 
attended by no appreciable capacity effects ; it is not 



POWER TRANSMISSION. 183 

subject to electric surgings, which sometimes cause insu- 
lation perforations, short circuits, and arcing. It permits 
the use of direct-current motors with their very satisfac- 
tory operation as to efficiency, small starting current, 
overload capacity, and speed control. Its use on trans- 
mission lines of over a few miles' length is prohibited by 
the cost of the line which it necessitates. As will be seen 
later, long distance electrical power transmission, to be 
economical or even commercially possible, must be ef- 
fected by high voltages. Direct-current sparkless com- 
mutation is limited to 1000 volts. This Hmit is dependent 
upon the economical and mechanical limits of armature 
peripheral velocity, current density, gap-flux density, and 
temperature elevation. Furthermore service conditions 
demand other voltages than those of the transmission line. 
The direct-current transformer or dynamotor is expensive 
and not very efficient. 

The use of alternating currents is attended by the evil 
effects of inductance and capacity ; the operation of alter- 
nating-current motors can be called only fairly satisfactory ; 
but the employment of the very satisfactory, highly effi- 
cient, and moderately priced static transformer, makes 
possible the transmission at high voltages with its accom- 
panying small currents, small line wires, and cheap pole 
line construction. 

The use of the synchronous converter for distribution 
purposes in connection with alternating-current transmis- 
sion, constitutes a very satisfactory system, and seems to 
best meet all the engineering requirements. 

87. Frequency. — It is customary to call frequencies 
above 60 high, and those below 60 low. The proper fre- 



l84 ALTERNATING-CURRENT MACHINES. 

quency for a transmission and distributing system is 
dependent upon a number of variables as follows : — 

a. High-frequency transformers are smaller and cost 
less than those of lower frequency. This is seen by inspec- 
tion of the formula in article 59, I. For the same volt- 
age and flux density, the product of the iron cross-section 
and the number of turns varies inversely as the frequency. 
The cross-section of copper would be the same for the 
same capacity, irrespective of the frequency. 

h. High-frequency generators may be constructed 
cheaper than those of low frequency. For the same field 
multipolarity a high frequency is associated with high arm- 
ature speed, and, therefore, greater output. On the other 
hand, if an armature be run at the greatest peripheral velo- 
city mechanically permissible, a high frequency necessitates 
a greater field multipolarity, and, therefore, a greater cost 
and com.plexity of construction. 

c. High frequencies permit of the satisfactory oper- 
ation of both arc and incandescent lamps. Arcs do not 
operate well on any frequencies below 40. The satisfac- 
tory operation of incandescent lamps depends upon their 
voltage and candle-power. Low-voltage lamps have fat 
filaments of large heat capacity which do not drop in tem- 
perature so rapidly as high-voltage thin filaments. The 
same is true of high candlepower filaments. These lamps 
may be operated satisfactorily at 25 cycles per second. 
Standard iio-volt, 16 candle-power lamps, however, fatigue 
the eye at frequencies under 30 cycles. 

d. The inductive line drop, 2 tt/Z, varies directly as the 
frequency. Its value will be considered later. Being 
greater for high frequencies, it is then more liable to pro- 
duce poor regulation at points of distribution. 



POWER TRANSMISSION. 185 

e. The capacity charging current also varies directly as 
the frequency. 

/. The wattless currents clue to inductance and capacity, 
therefore, increase with the frequency, and thereby lower 
the operative capacity of the generator, the transformers, 
and the line. They also lower the efficiency of operation. 

g. High frequencies may necessitate so high a field 
multipolarity that the angular speed variation of the prime 
mover will prevent the satisfactory paralleling of the gen- 
erators. For the same reason, the running of synchronous 
motors and of synchronous converters may be unsatisfac- 
tory. 

Ji. Induction motors are best suited for operation on low- 
frequency circuits. At high frequencies the speed must 
be high or the motor must be large to avoid running on a 
low-power factor. The speed could be lowered by increas- 
ing the number of poles ; i.e., by placing the poles nearer 
to each other. If the diameter remained the same, this 
would result in an increase of stator flux leakage, which 
would reduce the power factor. 

88. Voltage If the frequency, the amount of trans- 
mitted power, and the percentage of power lost in the line, 
remain constant, the weight of line wire will vary in- 
versely as the square of the voltage impressed upon the 
line. This depends upon the fact that the cross-section of 
the wire is not determined by the current density and the 
limit of temperature elevation, but by the permissible 
voltage drop. If the impressed voltage on a line be 
multiplied by n, the drop in the line may be increased n 
times without altering the line loss. For the Hne loss is to 
the total power given to the Hne as the drop in volts is to 



1 86 



ALTERNATING-CURRENT MACHINES. 



the impressed voltage. To transmit the same power, but 
-th the previous current is necessary ; and this current, to 

produce n times the drop, must, therefore, transverse a 
resistance 7i^ times as great as previously. 

In transmitting power electrically over long distances, 
the line cost constitutes a large part of the total invest- 

12,5 



10.0 



:5.o 



2.5 

































1 
















1 
















1 










\ 


"cm 

CM 


wOl 
















1 


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/ 


/ 


/ 










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iy 


y 










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20 



30 



40 
KILOVOLTS 
Fig. 142. 



50 



60 



ment. In such cases it is desirable to employ as high volt- 
ages as possible. There is, however, a limit to the voltage 
which may be employed. Mr. Charles F. Scott has given 
some interesting results of experiments carried out on vari- 
ous pole hues. He found that the power lost through the 
air between wires increased with the impressed voltage, and 
after a certain voltage was reached, increased very rapidly ; 
that, with a given impressed voltage, the loss decreased as 



POWER TRANSMISSION. 



187 



the distance between the wires was increased ; that atmos- 
pheric conditions, such as snow, rain, and humidity, had no 
appreciable effect on the loss ; that peaked wave-shaped 
E.M.F.'?> gave a greater loss than flat -topped ones ; and 
that the loss decreased as the diameter of the wires was 
increased. The relations between the distance between 
wires, the impressed voltage, and the power loss, is shown 
in Fig. 142. 



9 














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Fig 




OLTS 
143. 


5 





6 






The influence of the change of size of wire is shown in 
Fig. 143, where the distance between the wires was 48 
inches in both cases. The influence of the size of the con- 
ductor surfaces upon the voltage necessary to break down 
a dielectric can be illustrated by the apparatus shown in 
Fig. 144. An applied voltage of sufficient magnitude will 
produce a spark between pointed conductors, although the 



ALTERNATING-CURRENT MACHINES. 



path may be longer than between those which are spherical 
and are connected in parallel with them. 

At high voltages the leakage is accompanied by a hissing 
sound, and the wires glow visibly at night. 

The maximum pressure thus far employed in practice is 
60,000 volts. The Standard Electric Company of Califor- 




r\ 



o o 







Fig. 144. 

nia uses this voltage on a line constructed of aluminium 
cable I" inch in diameter, and made up of 37 strands, the 
different cables being 42'^ from each other. 

Very long distance electrical power transmission is most 
economically accomplished by the employment of such high 
voltages. As generators cannot be constructed to give 
much higher than 1 3,000 volts, and utilization devices are 
also limited as to the voltage which may be impressed upon 
them, step-up and step-down transformers are necessary. 

To produce the same percentage loss of power in a 
line when its length is varied, the impressed voltage must 
vary as the length. The number of volts per mile vary 
in practice from 300 to 2000. The choice of voltage is 
determined by balancing the annual value of the energy 
lost in the line against the interest and depreciation on the 
extra capital invested necessary to prevent the loss. 

As the distance of transmission decreases there arrives 



POWER TRANSMISSION. 189 

a point when step-up transformers can be dispensed with 
and also some step-down transformers. A further decrease 
of distance permits of transmission and distribution with- 
out the use of any transformers. 

89. Number of Phases A comparison of the weights 

of line wire of a given material, necessary to be used in 
transmitting a given power, at a given loss, over the same 
distance, must be based upon equal maximum voltages 
between the wires. For the losses by leakage, the thick- 
ness and cost of insulation, and perhaps the risk of danger 
to life, are dependent upon the maximum value. A com- 
parison upon this basis gives, according to Steinmetz, the 
following results : — 

Relative weights of line wire to trafismit equal power over the 
same distance at the same loss^ with imit power-factor. 

2 Wires. Single-phase loo.o 

Continuous current . . . . 50.0 

3 Wires. Three-phase 75.0 

Quarter-phase i45-7 

4 Wires. Quarter-phase loo.o 

The continuous current does not come into consideration 
because of its voltage limitation. The single-phase and 
4-wire quarter-phase system each requires one-third more 
wire than the three-phase system. 

By use of the Scott three-phase quarter-phase trans- 
former the transmission system may be three-phase, while 
the distribution and utilization system may be quarter- 
phase. 

90. Aluminium Line Wire There are but two mate- 
rials available for the construction of long-transmission 



190 ALTERNATING-CURRENT MACHINES. 

lines. The high permeabiUty of iron prohibits its use. 
The remaining materials are copper and aluminium. The 
prices of both metals vary, and sometimes it is cheaper to 
use one metal, and again to use the other. A number of 
aluminium lines have been constructed on the Pacific 
coast. Not all of them have proved satisfactory. Some 
of them broke very frequently and without apparent undue 
strain. Experience has shown that the troubles were due 
either to improper alloying or impurity of the material, or 
to improper stringing of the wires. Aluminium has a 
large temperature coefficient of expansion. Allowance 
should be made for this. The Standard Electric Co. 
strings so as to subject their aluminium cables to a strain 
of 4000 lbs. per square inch at 20° C. Perrine and Baum 
give the following data concerning a line of commercial 
aluminium in which they were interested : — . 

DATA CONCERNING ALUMINIUM. 

Size of Aluminium Wire = No. i copper. 
Resistance of Aluminium Wire = No. 3 copper. 
Tensile Strength of Aluminium Wire = No. 5 copper. 
Weight of Aluminium Wire = No. 6 copper. 

Diameter for the same conductivity 1.270 times copper. 
Area " " " " 1.640 times copper. 

Tensile Strength for the same conductivity 0.629 times 

copper. 
Weight for the same conductivity 0.501 times copper. 

91. Line Resistance. — The resistance of anything but 
very large lines is the same for alternatirrg currents as for 
direct currents. In the larger sizes, however, the resist- 
ance is greater for the alternating currents. The reason 



POWER TRANSMISSION. 



191 



for the increase is the fact that the current density is not 
uniform throughout a cross-section of the conductor, but 
is greater toward its outside. The lack of uniformity of 
density is due to counter electromotive forces set up, in 



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20 



30 



70 



80 



100 



40 50 60 

MILLIONS 

CIRCULAR MILS X FREQUENCY 

Fig. 145. 

the interior of the wire, by the varying flux around the 
axis of the wire which accompanies the alternations of the 
current. This phenomena is termed skin effect. Its 
magnitude may be determined from the curve in Fig. 145. 

92. Line Inductance. — The varying flux, which is set 
up between the two-line wires of a single-phase trans- 
mission circuit by the current flowing in them, gives rise 
to a self-induced counter E.M.F. The inductance per 
unit length of single wire is numerically equal to the flux 
per unit current, which links a unit length of the line. 
To determine this value consider a single-phase line, with 
wires of R cms. radius, strung with d cms. between their 
centers, and carrying a current i. Let a cross-section of 



192 



ALTERNATING-CURRENT MACHINES. 




The flux d^^, which 



Fig. 146. 

the hne be represented in Fig. 146. 

passes through an element dr wide and of unit length, is 

equal to the magnetomotive force divided by the reluc 

tance or . ■ 
d^. = 

~d^ 

Integrating for values of r between d 

'd-R 



R and R 



and practically 



$1=2/ log 



2 i log 



R 



There is some flux which surrounds 
the axis of the right-hand wire, and 
which lies inside the metal. This is of 
appreciable magnitude owing to the 
greater flux density near the wire. 
Represent the wire by the circle in 
Fig. 147, and suppose that the current 
is uniformly distributed over the wire. 

inside the circle of diameter x 




Fig. 147. 

Then the current 



is —2 i, and the magneto- 



motive force, which it produces, is 






POWER TRANSMISSION. 193 

The flux, however, which it produces, Hnks itself with 
but ^,ths of the wire. The flux througli the element dx, 
which can be considered as linking the circuit, is therefore 

2 x^idx 



Integrating for values of x between o and R, 

For copper or aluminium wires /x = i. Hence the total 
flux linked with the line is 



$1 + $2 = 2 2 



■*i|)*i 



and the inductance, in absolute units, being the flux per 
unit current, is 

This gives by reduction the inductance in henrys per wire 
per mile as 

d' 



Z = (^8o.5 + 74olog(^-jjio-l 

In case of a three-phase line, the inductance in henrys per 
mile of the whole circuit is 

^ = (139 + 1,280 logf^ 

93. Line Capacity The two wires of a single-phase 

transmission line, together with the air between them, 
act as a condenser. The wires correspond to the con- 
denser plates, and the air to the dielectric. When Hues 
are long, or when the wires are close together, the capacity 



194 



ALTERXATIXG-CURREXT MACHINES. 





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POWER TRANSMISSION. 

CURRENT IN MAIN CONDUCTOR. VALUES OF T. 



195 



System. 


Per Cent Power Factor. 


100 


•95 


.0 


•85 


.80 


Single-phase 

Two-phase (4 wire) .... 
Three-phase (3 wire) . . . 


1. 000 
.500 

.576 


1.052 
.526 
.607 


I. Ill 

•555 
.642 


1. 172 

.588 
.679 


1.250 
•625 
.729 



Current in main conductors 



Output in Watts 



X T. 



is quite appreciable. The capacity of a two-wire line in 
microfarads per mile of line is approximately 



C = 



0.04 



■-S) 



where d is the distance between centers of wires, and R is 
the radius of the wire, both being measured in the same 
units. 

Because of its capacity, a line which is unloaded takes 
a current when an alternating E.M.F. is impressed upon 
it. If the capacity be C microfarads, then E volts at a 
frequency/ would send a charging current (§21) 

/= 2 'KfEC\o~'^ amperes. 

94. Line Constants The various constants of a trans- 
mission line are given in the table on the preceding page. 

In calculating the sizes of lines, transformers, and 
generators of a transmission system, allowance has to be 
made for the various power factors of the load drawn off 
at various points. Induction motors, arc lights, and 
synchronous motors under some excitations have other 
than unit power-factor. Therefore transformers which 
supply them must have an excess of capacity sufficient to 



96 



ALTERNATING-CURRENT MACHINES. 



Line loss in per cent of power delivered 

28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 n 10 9 8 7 6 5 4 



3 2 1 



^100 
05 105 
|l10 

■=* 120 
^125 
^130 
135 
140 
145 
150 
155 
'160 
165 
170 
175 
180 
185 
190 
195 
200 



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Line loss in 



7161514 1312 11109 8 7 6 5 4 

per cent of power delivered 
Fig. 148. 



3 2 10 



POWER TRANSMISSION. 197 

carry the extra current. The hne, the step-up trans- 
formers, and the generator which suppHes the electrical 
energy, must all have increased capacity. The prime 
mover, which drives the generator, however, does not need 
to have this extra capacity. The actual current in all the 
apparatus being larger than it would be if the power- 
factor were unity, is accompanied by increased heat losses 
at every point. The excess of capacity is needed to get 
rid of this heat, without undue elevation of temperature 
in the apparatus. The equivalent impedance of the loads 
and their equivalent power-factor as affecting the line can 
be determined as shown in the problems of Chapter IV. 

95. Weight of Copper In the curves of Fig. 148 are 

shown the relations which exist between the transmission 
loss of power in per cent, the impressed volts per mile, and 
the weight of copper per k.w. delivered. The loss is 
expressed as a percentage of the power delivered. The 
curves apply to a three-phase transmission at unit power- 
factor. Five per cent has been allowed for sag of lines be- 
tween poles. To determine the values for aluminium wire, 
correct by the constants given in § 90. 



198 ALTERNATING-CURRENT MACHINES. 



CHAPTER X. 

TESTS. 

96. Apparatus In the following pages are given direc- 
tions for a series of experiments designed to give the stu- 
dent dexterity in handling apparatus, a firmer grasp of the 
phenomena connected with alternating currents, and a 
knowledge of the methods employed in testing alternating- 
current apparatus. This course was laid out for use in 
a laboratory with but a moderate amount of apparatus, 
and all this apparatus will be here described to avoid the 
necessity of introducing such descriptions in the directions 
for the experiments. 

The laboratory is supplied with power from an Edison 
direct-current three- wire system with 117 volts on a side. 
The largest machine is a 7.5 k.w. double-current gener- 
ator, which is run as an inverted converter from the Edi- 
son current. This is a four-pole machine whose speed can 
be regulated from 1200 to 1800 r.p.m. This gives a 
range on the alternating end of 40 ^ to 60 ^. There are 
six slip-rings on the armature, so connected that single- 
phase current can be had from rings i and 4, quarter-phase 
from 1-4 and 2-6, and three-phase from 1-3-5. The volt- 
age, of course, cannot be altered. A laboratory not sup- 
plied with current from a street service could use such 
a machine, running it as a double-current generator by a 
steam or gas engine. This would be more desirable than 
running a regular alternator ; as frequently direct current, 



TESTS. 199 

as well as alternating, is called for in the experiments. In 
such case, both frequency and voltage could be regulated. 
Besides this, there is a 5 00- watt, 8-pole, I2 5rw alternator 
belt-driven by a direct-current motor. The wave-shape of 
this machine was given in Fig. 4. The machine on which 
most of the tests are run is a double-current generator of 
about I K.w. capacity. This is a bipolar machine fitted 
with four slip-rings on one end and a commutator on the 
other. The rings are arranged so that three-phase cur- 
rent is obtained from rings 1-2-3, ^i^d single-phase from 
rings 1-4. This machine serves a multitude of purposes. 
It can be run as a direct-current motor ; as a synchronous 
motor, either single-phase or three-phase ; as a converter, 
either direct or inverted ; and when driven by a belt, as an 
alternator, single-phase or three-phase ; or as a direct-cur- 
rent generator, either shunt wound or separately excited. 
Its speed can be varied from 1500 to 2400, giving fre- 
quencies of 25 ^ to 40 .w. It may be run in parallel 
with the larger converter when that is slowed down to 
40 ^. The equipment of rotating apparatus is completed 
by two induction motors, one of one-horse power, the other 
of a half-horse power capacity. They are both wound for 
three-phase ; but the smaller is equipped with a condenser 
compensator, as described in § 6j, and can be run when 
desired on a single-phase circuit. 

The transformer equipment consists of three i-k.w. 
I to I oil-cooled transformers, a half-K.w. i to 2 air- 
cooled transformer, and an old ring-wound armature ar- 
ranged with taps so that it can serve to transform from i 
to I, 2, 3, or 4. 

For inductive circuits three coils are used. The first, 
known as Coil i, was described in § 9. It has about 3000 



200 ALTERNATING-CURRENT MACHINES. 

turns of No. i6 B. & S. wire, lo ohms resistance, and 0.2 
henrys inductance without iron. A bundle of iron wires 
\6" long and \\" diameter can be inserted in either of the 
three coils. Coil 2 is in the shape of a hollow cylinder, 
whose external diameter is ^\" , internal diameter 2\" , and 
length 3^'^ It consists of about 6500 turns of No. 26 
B. & S. wire, with an inductance of o.i henry and a resist- 
ance of 60 ohms. Coil 3 is of the same external appear- 
ance as Coil 2, but is made of about 7600 turns of No. 25 
B. & S. wire, giving an inductance of 0.141 henry and a 
resistance of 60 ohms. It will be noticed that Coil 2 and 
Coil 3 have the same resistance, and that their inductances 
are as i to V2. Six paraffined paper condensers of about 
two microfarads each, are used when condensive circuits 
are desired. 

The instruments used are as follows : Four hot-wire 
ammeters, with ranges of i, 3, 15, and 20 amperes respec- 
tively. All but the first work across shunts, the small one, 
however, taking the whole current through its hot wire. 
These, of course, are used for either alternating or direct 
currents. Two inclined coil ammeters have ranges respec- 
tively of 5 amperes and 50 amperes. 

There are three voltmeters, an inclined coil instrument 
reading to 65 volts ; a Cardew hot-wire instrument, read- 
ing to I 50 volts; and a Weston standard portable voltmeter 
with two scales, one up to 100 volts, the other up to 200. 
Any of these may be used on either alternating- or direct- 
current circuits. 

For all the larger measurements a 2.5 k.w. indicating 
wattmeter is used. For the finer measurements a Weston 
standard wattmeter, having two scales, is used. The lower 
scale, for use with pressures of 75 volts or less, reads up 



TESTS. 201 

to 75 watts; the upper scale, for use with pressures of 150 
volts, reads up to 150 watts. For this instrument a shunt 
has been constructed, having a coil similar to the current 
coil of wattmeter, so as to have the same resistance and 
the same time constant as the latter. This is placed in 
parallel with the current coil, a small resistance for ballast 
having first been placed in series with each. The watt- 
meter then reads up to 300 watts, and is as accurate on 
inductive as on non-inductive loads. 

Certain direct-current instruments are occasionally used, 
principally a Weston standard portable 150-volt direct- 
current voltmeter, and a similar five-ampere ammeter. 
These instruments are used for convenience, and could be 
dispensed with if necessary. 

A means of measuring the rate of rotation of the various 
machines is essential, and a portable tachometer is by 
far the best instrument for the purpose. Of course, a 
greater accuracy can be obtained by using a revolution 
counter, and noting the number of revolutions in a con- 
siderable length of time ; but this method is too slow to 
be satisfactory, and is useless if the speed be fluctuating. 

To load a machine electrically, two lamp boards are 
used. These have each ten key sockets arranged in two 
rows between three wires. Thus, three wires of a three- 
wire system may be connected thereto, or the outside 
wires may be connected together, and all the lamps be put 
in multiple ; and finally, by using the two outside wires 
only, all the lamps turned on in one row can be put in 
series with all those on in the other row. Thus a wide 
range of resistances can be obtained by very small steps, 
if a few each of 8, 16, 32, 50, and 100 candle-power lamps 
are in the sockets. 



202 



ALTERNATING-CURRENT MACHINES. 



K 



In the following descriptions of experiments, for the 
sake of brevity, the apparatus needed will not be named ; 
but such notation will be used in the figures, showing the 
arrangement of apparatus, that the particular apparatus 
will be indicated. All measuring instruments will be 
marked with a letter indicating their kind, and a number 
indicating their capacity ; thus A.^ is a three-ampere am- 
meter, ^500 is a 2.5 K.w. wattmeter. In many cases, the 
manner of drawing will indicate the apparatus, thus : 



is an alternating-current ammeter or voltmeter. 

is a direct-current ammeter or voltmeter. 

is a wattmeter, the binding posts of the current coil 
being conspicuously large to avoid confusion. 

is a switch designed to shift one ammeter out of cir- 
cuit and another in without interrupting the con- 
tinuit}^ of the circuit. 

is a contact-maker, giving a short contact at any 
desired point in a revolution. 

is a commutator designed to change the direction of 
current flow in a circuit. 

is a lamp board as described above. 

is an inductive coil. 

is a condenser. 

is a transformer, the numbers indicating the relative 
number of turns. 

is the armature and field coils of a direct-current 
machine or the direct-current end of a converter. 

is the armature and field coils of an alternating-cur- 
rent machine or the alternating-current end of 
a converter. 

is to represent a belt-drive between two armatures. 

is to represent a direct connection, or, in the case of 
a converter, the two ends of the same armature. 



_i^m_ 



G — 



TESTS. 



203 



97. Exp. I. Peculiarities of Alternating-Current Circuits. 

— This experiment consists of some merely qualitative 
observations calculated to illustrate to the student the dif- 
ference between alternating currents and the direct cur- 
rents he has hitherto used. 

First Part. — Arrange the apparatus as in Fig. 149, the 
lamp being by way of protection, in the case of accidental 
short circuit. Let x be first, too ,c. p. 

the inductive coil known as W 

Coil I, second, the same with 
the iron core inserted in it, 
third a condenser of about 
10 M.F. capacity, and fourth 
a 50-candle power lamp. 
Apply to these circuits a 



60 v.« 
A. c. 



To X. 



Fig. 149. 



uniform potential of about 60 volts. Let the frequency 
be successively 125 ^, 40 ^, and o ^, i.e., direct current. 
With each change note the ammeter reading. It will be 
observed that with an inductive circuit the current in- 
creases as the frequency decreases, and that the maximum 
current possible flows in the form of direct current. With 
a condensive circuit the current decreases as the frequency 
decreases, and is zero with direct current. With a non- 
reactive circuit, such as 
^—\ " ' — r-.. the 50 c. p. lamp, the 

current flow is indepen- 
dent of the frequency. 
Direct current at 60 
volts for this experiment can be secured, of course, by 
running a small dynamo at suitable excitation, but more 
easily from the 1 1 7-volt street service by the arrangement 
shown in Fig. 150. The lamps can be adjusted to give 



117 V, 

D. c. 



W\ 



50 16 

50 



-wXy 



60 y. 



Fig. 150. 



204 ALTERNATING-CURRENT MACHINES. 

60 volts, and the rheostat can take up the difference. 
This adjustment will be, of course, somewhat different for 
different loads. 

Second Part. — The following solution is one of the 
many used for blue-prints, and has, besides, the property of 
turning blue, at the anode only, when a current is passed 
through it, if the anode be of iron. Mix 25 parts (by 
weight) of ammonium nitrate, NH_^ NO3, and 12.5 parts of 
ammonium muriate, XH^Cl. Dissolve 1.3 parts of ferri- 
cyanide of potassium, K^Fe (CX)^., (red prussiate of potash) 
in 1000 parts of water. Add the ammonium salts. The 
chemicals should be pure and the water distilled. Keep 
in a dark place, and use within twenty-four hours. 

Prepare an insulating handle, Fig. 151, with three piano- 
wire projections long enough to be elastic, and whose 
points may touch a plane surface in' a right line and 



16 c. p. 



75 V. 
3 Phase ^ 
40 ^ 




Fig. 151. 

near together. Let these wires be connected through 16 
c. p. lamps respectively to the terminals of a three-phase 
system, the pressure being 100 volts or less, and the fre- 
quency 40 or less. Lay an uncalendered paper well 
moistened, but not soaked, in the blue-print solution upon 
a metal plate, and draw the marking-points quickly across 
its surface. Blue marks will be left when the current is 
passing in one of its directions ; and these, by their inter- 
rupted nature, will show the change of direction in the 
alternating current. Also the relative displacement or 



TESTS. 



205 



240 
210- 



o£ the rows of marks will show 
the phase displacements of a three-phase 
current, as in Fig. 152. 

TJiird Part. — Excite a i6-candle power 
lamp with alternating current at its rated 
voltage and a low frequency, say 40 ^. 
Hold one end of a bar magnet against the 
bulb and in various positions. The fila- 
ment will vibrate synchronously with the 
alternations, due to the regularly recurring 
attraction and repulsion between the per- 
manent magnetic field of the magnet and 
the alternating field of the filament. If 
this experiment fails at first, try varying 
the frequency, the strength, and polarity 
of the magnet, and even try other lamps. 
Often the filament can be made to so vi- 
brate as to touch the glass, and finally 
rupture itself. 



98. Exp. 2. Shape of E.M.F. Wave of 

Alternator To perform this experiment 

use is made of a balance, as shown in 
Fig. 153. It consists of a hard graphite 
rod, C, of high resistance, through which 
current is passed from a direct-current con- 
stant potential source, two 16 c. p. lamps being in series 
to guard against accident in case of accidental short cir- 
cuit. A rolling contact bears upon this rod, and allows 
of a nice adjustment of the pressure applied to the testing 
circuit. This pressure can be accurately measured by the 
standard direct-current voltmeter V. In one branch of 



Fig. 152. 



206 



ALTERNATING-CURRENT MACHINES. 



the test circuit is placed the telephone receiver, T, The 
operation is as follows : The test circuit, consisting of the 
armature of the alternator, a lamp or other non-inductive 
resistance for protection, a contact-maker, and the E.M.F. 
balance just described, is closed for an instant at some 
point of the revolution which corresponds to some point 
of the curve of instantaneous pressures. At such instants 
current will flow through the test circuit, causing the 
telephone receiver to click sharply ; and this click comes 

with a rapidity corresponding 
to the rate of revolution of 
the contact-maker, say 1800 
per minute. The sliding con- 
tact on the graphite rod is 
then operated until the con- 
tinuous direct. E.M.F. is just 
equal and opposite to the in- 
stantaneous E.JM.F. put forth 
by the alternator. Then there 
will be no flow of current 
whether the contact-maker be 
^^* ^^^' opened or closed, and the re- 

ceiver will cease to click. The voltage can be read di- 
rectly on the voltmeter, thus obviating the use of any 
reduction constants. This method, due to Mershon, is 
very delicate, since a telephone receiver is sensitive to 
very small currents. 

To obtain an E.M.F. curye from an alternator, arrange 
the apparatus as in Fig. 154. The contact-maker is con- 
nected directly to the shaft of the generator, and is obliged 
to revolve in unison therewith. Run the alternator at its 
rated speed and voltage. Set the brush of contact-maker 




TESTS. 



207 



at the desired beginning point, and balance the instanta- 
neous E.M.F. by sHding the balance until no clicking is 
heard in the receiver. Note the setting of the contact- 




iSJ ^G)- 



16 c.p. 



To Balance 



Fig. 154. 

maker and the reading of the voltmeter in the balance. 
Set the contact-maker ahead by five electrical degrees 
( I mechanical degree — p electrical degrees, where / is the 
number of pairs of poles), and repeat as before. Take 
readings thus by steps of five degrees throughout one 
complete cycle, i.e., under two poles. Since the instan- 
taneous E.M.F. will be in one direction during half a 
cycle, but in the opposite direction during the other half, 
and balancing E.M.F. is always in the same direction, 
a commutator must be introduced in the test circuit, as 
shown. When the commutator is in one position, the 
voltage readings should be marked -f , when in the other 
position, they should be marked — . 

Plot a curve with volts as ordinates and degrees as 
abscissae. Indicate on the margin of the curve-sheet the 
effective value of the curve as obtained from the alternat- 
ing-current voltmeter. By means of a planimeter meas- 
ure the area of one lobe of the curve, and find its average 
ordinate, by dividing the area by the base line, i.e., the 



2o8 



ALTERNATING-CURRENT MACHINES. 



length corresponding to i8o°. This may be done in inch 
and square inch units, if the planimeter be so caUbrated, 
without reducing. Lay this average ordinate off on the 
margin also. Divide the effective value by the average 
value to obtain the form-factor (§ 4) of the pressure wave. 
If this value be about i.ii, the curve is practically a 
sinusoid. 

99. Exp. 3. Shape of Current Wave of Alternator with 
Inductive Load. — Arrange the apparatus as shown in 
Fig. 155. The method of procedure is that of Exp. 2. 

The instantaneous drop of potential along a non-induc- 
tive resistance is proportional to, and in phase with, the 
current in that resistance. Measure the resistance of 



Coil 1 




To Balance 



the 50 c. p. lamp under tJie coiiditions of use, since the 
resistance of a carbon filament varies widely with the 
temperature ; and from this, and the values of instanta- 
neous pressures observed, calcujate the instantaneous cur- 
rents according to Ohm's Law. 

Plot a curve with amperes as ordinates, and degrees as 
abscissae. 



TESTS. 



209 



100. Exp. 4. Simultaneous Pressure, Current, and 
Power Curves from Alternator with an Inductive Load. — 

Arrange apparatus as in Fig. 156. It will be seen that 
either a point on the pressure curve (Exp. 2), or a point 
on the current curve (Exp. 3), can be taken by suitably 
placing the two-throw switch. Putting the commutator 
in the main circuit instead of the test circuit is possible 
when the load is light, does not affect the validity of the 
observations, and eliminates a possible source of trouble 
from bad contacts in the test circuit. Readings are to be 




p.,D.r. 

Switch 



Fig. 156. 

taken every five degrees through 400°, a little over one 
cycle. Take readings for both pressure and current curves 
each time before moving the contact-maker. This is better 
than taking a complete current curve, then going back and 
taking a pressure curve, since there is more liability to 
distortion due to change in conditions in the latter case. 
The voltmeter, ammeter, and wattmeter readings should 
not vary during the test, and occasional observations should 
be made to see that this condition is complied with. If it 
cannot be, readings at stated intervals should be taken, 
and their averages used in the subsequent calculations. 



2IO 



ALTERNATING-CURRENT MACHINES. 



Plot three curves on one sheet, having degrees as their 
common abscissae, and volts, amperes, and watts as their 
respective ordinates. The instantaneous watts at any 
abscissa equal the product of the instantaneous volts and 
amperes for that same abscissa. In general, a separate 
scale of ordinates will be required for each curA'e. The 
curves will have the general relations shown in Fig. 14. 

Note the number of degrees intercepted on the axis, 
between the pressure curve and the current cun^e. This 
is the angle of lag, ^, the cosine of which is the power- 
factor of the circuit if the pressure wave is sinusoidal. 
By the method given in Exp. 2, find the form-factor of the 
pressure curve. Divide this by the form-factor of a true 
sinusoid, i.e., i.i i, and call the quotient K. Then K cos <h 
is the power-factor of the curve, whether sinusoidal or not. 

By means of a planimeter, measure the area of the lobes 
of the power curve, being careful to go around the nega- 
tive part in a counter-clockwise direction. Find the mean 
ordinate of this curve by dividing the area b}' the base 
line, and determine its value in watts by laying off on the 
scale of ordinates for the power curve. 

Fill out the following table, putting in the last column 
the percentage variation of the individual values from the 



How Determined. 


Watts. 


Tc 


By Wattmeter 






By Planimeter 






By E y. I y. K cos . . . . 






Average 






TESTS. 211 

The variations should be within the limits of errors of 
instruments and observations, say 2%. 

loi. Exp. 5. Measurement of Self -inductance. — There 
are various methods of measuring the coefficient of self- 
induction, two of which are given here. The first is 
applicable to any series circuit, and consists in the deter- 
mination of the quantities in the general expression 



Vi?^ -h (2 7r/Z)2 

and solution for L. If E, I, and R be measured respec- 
tively in volts, amperes, and ohms, L will be expressed 
in henrys. 

ia) Arrange apparatus as shown in Fig. 157, all the 
lamps being turned off. Insert at x successively Coil i. 
Coil 2, and Coil 3. Turn 
on lamps until a good 
ammeter deflection is ob- 
tained, and note readings 
of ammeter and volt- 
meter. The ohmic re- 
sistance must in each ^^' ^^^' 
case be independently determined if not already known. 
Take four sets of observations with each coil; with and 
without iron core, at 40 ^, and 60 ^. 

Solve for the inductance in each case from 

2 77/ 

Without iron in the magnetic circuit, Z is a constant of 
the circuit, independent of / and/; but when iron is pres- 




212 



ALTERXATIXG-CURREXT MACHINES. 



ent, it varies considerably ^^ith / and slightly with f. 
The variation of inductance with load is the subject of 
Exp. 7. 

Caution must be used in this experiment, that the am- 
meter be not injured. For instance, the careless removal 
of an iron core with closed circuit may cause a destructive 
increase of current. 

{b) The above method of measuring the inductance is 
not applicable to branched or parallel circuits with different 
time constants, for the reason that the resistance of the 
whole circuit, as measured by direct current, is not the 
equivalent resistance of the circuit, as explained in § 28. 
A method using a voltmeter, ammeter, and wattmeter is 
entirely general, is equally accurate, and does not require 



A 



A, 



120 V. 
A.C. 



3^ 



TIM 



060 



ToX. 



Fig. 158. 



the independent determination of the resistance. Arrange 
the apparatus the same as in the first part of this experi- 
ment, with the addition of a wattmeter, as shown in 
Fig. 158. The method of procedure is the same as be- 
fore, save that the wattmeter readmg is also noted in each 
case. If /, E, and P be the instrimient readings in am- 
peres, volts, and watts respectively, then the inductance in 
henrvs is 



E . 
— sin 



i'°'~'^) 



2^/ 



TESTS. 213 

This equation results from a consideration of the fol- 
lowing : — 

P 
cos <^ = ;^' 



L = Z sin 4> (see Fig. 30). 
J sin (cos- ^) 



.-. Z 



102. Exp. 6. Measurement of Capacity When there 

is no resistance and no inductance in a circuit — as is the 
case with a condenser — the general formula 



\/^^ + ("^-;^J 



reduces to /= wCfi", 

hence ^_ / 



2 7r/e 

Arrange the apparatus as in Fig. 159. Let x be the six 
condensers taken, first two at a time, then three at a time, 
then all together, always arranging 
them in parallel. ■ Note the current 
and pressure in each case, and i25'N.a. 



^ 



solve for C by the above formula. 
The capacity of any parallel com- 
bination of condensers is the sum ^s- 159. 
of the capacities of the component parts, and should so be 
shown by this experiment. If £ and / be in volts. and 



214 ALTERXATIXG-CURREXT MACHINES. 

amperes respectively, then C will be in farads. In the 
report reduce these results to microfarads by multiplying 
by io'\ 

This method is not open to the objections to the similar 
method of measuring inductance, since here the resistance 
is practically zero. Yet the second method, using the 
wattmeter, could be employed. The formula would be 

-L=Zsin(cos-^ 

But the wattmeter will read zero, since little power is lost 
in a condenser, so <^ = 90°, and 

or C=^, 

which is the same as deduced from the general formula. 

103. Exp. 7. Variation of Coefficient of Self -Induction 

Under Load This experiment may be performed in two 

parts ; (^7) by varying the magnitude of the measuring 
current, {b) by using a constant measuring current, and 
varying the saturation of the magnetic circuit b}- a separate 
current in a separate winding. 

{a) Measure the coefficient of self-induction of the fine 
wire coil of a i to 2 transformer by either of the methods 
of Exp. 5. This current must be made to vary by suitable 
steps, and this can most easily be done by applying differ- 
ent pressures to the coil. A wide assortment of pressures 
can be obtained by using different brushes of the converter 
supplying the energy, and the different steps of the 
I, 2, 3, 4 transformer. Determine the value of Z for each 



TESTS. 



215 



of the conditions, and plot a curve having these values as 
ordinates, and the corresponding currents used in measur- 
ing as abscissae. A curve, such as Fig. 160, will result 



.341 




.5 1 1.5 2 2.5 

MAGNETIZING CURRENT. AMPERES 
Fig. 160. 



3.5 



with certain irons if the current be started low enough. 
The sharp rise of the curve at first is due to the fact that 
at very low densities the permeability increases with density, 
as is shown in the curves on page 24, Vol. i. 




Fig. 161. 

{b) Arrange the apparatus as in Fig. 161. The meas- 
urements of L are made on the fine-wire side of the 1:2 



2l6 ALTERNATING-CURRENT MACHINES. 

transformer, while the permeabihty is altered by direct 
current in the low-pressure side. The measuring current 
should be kept constant ; and as it has a tendency to rise 
as L decreases, resistance will have to be inserted in the 
alternating-current circuit by adjusting R^^. Take read- 
ings at suitable steps from zero amperes direct current to 
the maximum safe temporary ampere capacity of the coil 
in question, say 15 amperes for a i^ k. w. 5 5 -volt coil. 

Calculate the value of L for each of the steps, and plot 
a curve, using these values as ordinates and the direct- 
current magnetizing amperes corresponding thereto as 
abscissae. 

104. Exp. 8. Measurement of Mutual Induction 

Arrange the apparatus as in Fig. 162, the requisite pres- 

Coil 2 Coil 3 



• •v^_>» • I I 






160 V.*- 
60~' 

Iron core through both 
Fig. 162. 

sure being secured by stepping up in the i : 2 transformer. 
The experiment consists of three parts : — 

{a) Determine the coefficient of mutual induction be- 
tween the two coils from the formula 

Transpose Coil 2 and Coil 3, and determine J/ again, the 
formula changing, of course, to 

E, = (0J//3. 

The results should be alike if the same current flows 
in each case. 



TESTS. 217 

Finally calculate the theoretical value of M on the as- 
sumption of no magnetic leakage from 

M = VAZ3. 

Zg and Zg were determined in Exp. 5. If the same meas- 
uring currents be used throughout, this last value of M 
will be somewhat above the others, since there is some 
leakage flux. 

{b) With the arrangement of Fig. 162 place the iron 
core with its end flush with the outside of Coil 2, and pro- 
jecting clear through Coil 3. Move Coil 3 by steps of 
2 cm. each from o to 24 cm. from Coil 2, and measure 
the value of M ior each step. Be careful that the iron 
core be not moved relatively to Coil 2. 

Plot a curve with centimeters as abscissae and values of 
J/ as ordinates. 

(c) Repeat the last, keeping the iron flush with Coil 3 
however, and moving Coil 2. In this case the current in 
Coil 2 will vary, and the curve will be distorted by the 
effects of load and saturation as investigated in Exp. 7. 

Plot curve of (b) and that of {c) on the same sheet and 
to the same scale. 

Be careful not to remove the iron core entirely from the 
coil that is carrying the current, or the current will exceed 
the capacity of the ammeter. 

105. Exp. 9. Measurement of Power in a Single-phase 
Inductive Circuit. — There are various ways of measuring 
power in alternating-current circuits besides using a watt- 
meter, but none are as satisfactory. 

In the following it is desired to measure the power in 
Coil I : — 

(a) By the three-voltmeter method. Arrange the appa- 



2l8 



ALTERNATING-CURRENT MACHINES. 



Ai 



60 y. 

60'^ 



rOr 



Coil 



50 



Fig. 163 

sures indicated. The power, P, in the coil is 

I 



ratus as in Fig. 163, 
the non-inductive 
lamp resistance, R, 
having been previ- 
ously determined. 
With a lOO-volt al- 
ternating-current volt- 
meter note the pres- 



^-^(^^ 



£,'-K^). 



{b) By the three-ammeter method. Arrange the appa- 
ratus as in Fig. 164. If / be the reading of A.^, I^ that of 
A^, and /, that of A^, the power in the circuit is 



2 



/r - //), 



where R is the non-inductive resistance of the lamps, 
which must be independently determined for the condi- 
tions of operation. 



( 




r= 


^4^- 


ft" 










60 V. 
60 ~ 










Fig. 



Fig. 165. 



(c) By the combined method. Arrange apparatus as in 
Fig. 165. If /be the reading of A., /^ the reading of A^, 
and B the reading of the voltmeter, then the power in the 



con IS 



2 



/- 



RV 
R ' 



R- 



TESTS. 



219 



If it be desired to compare the results of (^), (^), and (c), 
arrangements must be made so that the same difference of 
potential may be applied to the terminals of Coil i in each 
case. 

These methods are rather impractical, and open to the 
two serious objections that a small error of observation 
may lead to a serious error in the result, and that the 
maximum accuracy can only be obtained when about as 
much power is consumed in the auxiliary devices as in the 
circuit under test. 



106. Exp. 10. Measurement of Power in Polyphase 
Circuits by Indicating Wattmeters In any two-phase cir- 
cuit of four wires the load can be measured by two watt- 
meters, one connected regularly in each phase. The sum 
of their readings is the power in the circuit. In a two- 
phase four-wire system with a balanced load, one of the 
wattmeters may be dispensed with, and the reading of the 
other multipHed by two. 

In any two-phase, three-wire system the power can be 
measured by two wattmeters connected as in Fig. 166. 




Fig. 166. 



The sum of the instrument readings is the whole power. 
In a two-phase, three-wire system, where all the load is con- 



220 



ALTERXATIXG-CURREXT MACHINES. 



n 






n. 




nected between the outside wires and the common wire, 
and none between the outside wires themsekes, and where 
the load is balanced, then one wattmeter can be used to 
measure the whole power by connecting its current coil in 
the common wire nd its pressure coil between the com- 
mon wire and one outside wire first, then shiftinsr this 
connection to the other outside wire, as indicated in Fig:. 
167. The sum of the instrument readings in the two 

positions is the whole 
power. A wattmeter 
made with two pres- 
sure coils could have 
one connected each 
way, and the instru- 
Fig. 167. ment would automat- 

ically add the readings, gi\'ing the whole power directly. 
Or, again, a high non-reactive resistance could be placed 
between the two outside wires and the pressure coil of 
the wattmeter connected between the common wire and 
the center point of this resistance. This requires that the 
wattmeter be recalibrated with half of this hio^h resistance 
in series with its pressure coil. 

With the exception of the two-phase systems, the 
power in any balanced polyphase system may be measured 
by one wattmeter whose current coil is placed in one wire, 
and whose pressure coil is connected between that wire 
and the neutral point. The instrument reading multiplied 
by the number of phases gives the whole power. The 
neutral point may be on an extra wire, as in a three-phase, 
four-wire system ; or may be artificially constructed by con- 
necting the ends of equal non-reactive resistances together, 
and connecting the free ends one to each of the phase wires. 



TESTS. 



221 



With the exception of the two-phase systems, the 
power in any //-phase, ;/-wire system, irrespective of bal- 
ance, may be determined by the use of n—i wattmeters. 
The cm-rent coils are connected, one each, in ;/ — i of the 
wires, and the pressure coils have one of their ends con- 
nected to the respective phase w^ires, and their free ends 
all connected to the ;/th wire. The algebraic sum of the 
readings is the power in the whole circuit. Depending 
upon the power factor of the circuit, some of the watt- 
meters will read negatively, hence care must be taken 
that all connections are made in the same sense ; then 
those instruments which require that their connections be 
changed, to make them deflect properly, are the ones to 
whose readings a negative sign must be affixed. 

Some specific connections for indicating w^attmeters in 
three-phase circuits are shown in the following figures. 
Fig. i68 shows the connection of three w^attmeters to meas- 




ure the powder in an unbalanced three-phase system. All 
the readings will be in the positive direction, and their 
sum is the total power. If a fourth, or neutral wire be 
present, it should be used, instead of creating an artificial 
neutral, as shown. The magnitude of the equal non- 
reactive resistances, used to secure this neutral point, 



222 



ALTERNATING-CURRENT MACHINES. 




Fig. 169. 



must be so chosen that the resistances of the pressure-coils 
of the wattmeters will be so large, compared thereto, as 

not to disturb the po- 
r*~~*~l tential of the artificial 

neutral point. 

Fig. 169 shows the 
connection of one watt- 
meter, so as to read 
one-third of the whole 
power in a balanced, 
three-phase, four-wire 
system. If the system be three-wire, a neutral point 
may be created as in Fig. 168. 

Fig. 170 shows the connections of two wattmeters for 
the determination of the power in balanced or unbalanced 
three-phase systems, avoiding the necessity of a neutral 
point. The algebraic 
sum of the instru- 
ment indications is 
the whole power. If 
the power-factor be 
greater than .5, both 
instruments will give 
positive readings ; if 
it be less, one instru- ■^^^- ^7°- 

ment will give a negative reading. With low power-factors, 
such as given by a partially loaded induction motor, it is 
sometimes difficult to determine whether the smaller read- 
ings are negative or not. If in doubt, give the wattmeters 
a separate load of lamps (power-factor = i) and make the 
connections such that both instruments deflect properly. 
Then connect them to the load to be measured. If the 




TESTS. 



223 




terminals of one instrument have to be exchanged, then to 
the readings of that instrmiient must be affixed the nega- 
tive sign. Fig. 171 shows the connections for one watt- 
meter in a balanced three-phase circuit, independent of a 
neutral point. The free end of the pressure coil is con- 
nected first to one of the wires opposite that in which the 
current coil is con- 
nected, then to the 
other. The alge- 
braic sum of the 
readings in the two 
positions is the total 
power. Both read- Fig- 171. 

ings wdll be positive if the power-factor is greater than 
.5 ; but one of them will be negative if it is less. 
Hence care must be used to avoid confusion of signs at 
low power-factors. This method, requiring a two-throw 
switch to change the connection, tw^o readings of the 
instrument, and, if used on a load varying from high to 
low pow'er-f actor, a commutator, to change the pressure 
coil connections, has little advantage over the method of 
Fig. 169, save that it dispenses with the necessity for a 
neutral point. 

Six-phase circuits are used generally only between the 
step-down transformers of three-phase transmission sys- 
tems and the alternating-current ends of rotary conver- 
ters ; hence they are always balanced. They can then be 
measured by the method of Fig. 169, where a neutral is 
employed, or the three alternate wires may be considered 
a three-phase system, the method of Fig. 171 employed, 
and the three-phase power thus determined multiplied by 
2 to give the total power. If the circuit should be unbal- 



224 ALTERNATING-CURRENT MACHINES. 

ancecl, five instruments would be necessary, as stated earlier 
in this section. 

The student is expected to construct circuits according 
to the various figures just given, and convince himself 
that the wattmeters do give the true power. If the load 
be of lamps, the power in each may be measured by a volt- 
meter and ammeter used at their terminals ; then by con- 
necting in star and in delta, balanced' and unbalanced, the 
accuracy of the wattmeter indications can be checked. 

In following Fig. i66 or Fig. 167, it should be remem- 
bered that a two-phase current cannot be secured from an 
armature with a mesh winding, such as a rotary converter 
must have ; and that any attempt to make a two-phase, 
three-wire system out of a quarter-phase system will be 
disastrous. To get two-phase current from such a ma- 
chine, the quarter-phase current must be passed through 
the primaries of two similar transformers, two opposite 
wires going to one, the other two to the other. The 
transformer secondaries will then deliver true two-phase 
current, and the circuits may be united in a three-wire 
system. 

107. Exp. II. Calculation and Measurement of the Re- 
sulting Impedance of a Number of Impedances in Series. — 
Arrange apparatus as shown in Fig. 172. Determine the 
impedance Z of the whole circuit from the readings of the 
voltmeter and ammeter. 

--^^ 

Independently determine the ohmic resistance of the cir- 
cuit with the same current flowing. The magnitude of 
the current affects the resistance of the lamp. 



TESTS. 



225 



Solve for the reactance, X =2 -nfL, from the equation, 



Determine the angle of lag </> from 

reactance 

tan d) = — -. 

resistance 



4> 



tan 



X 



Determine the reactance, resistance, and impedance of 
Coil I, Coil 2, and the lamp, individually. The first two 
can be derived from the data of Exp. 5 without further 
measurements. 

Graphically determine the total reactance, resistance, im- 
pedance, and angle of lag by combining the individual parts 
in a parallelogram of impedances as described in § 26. A 
convenient scale for the actual plotting for a drawing-board 
2^" X 30'' is 2 ohms= I cm. 

Make a report in the form of a table such as the follow- 
ing. The variation, with careful work and good instru- 
ments, should not exceed 2%. 





Determined 


% 
Variation. 


Graphically. 


Experimen- 
tally. 


R . . . 








X . . . 








z . . . 








. . . 









226 



ALTERNATING-CURRENT MACHINES. 



120 V. 
40 ~' 




100 c. p. 



C-oil 1. 



Co;i 2. 



Fig. 172. 



108. Exp. 12. Calculation and Measurement of the Re- 
sulting Impedance of a Number of Impedances in Parallel. 

— Use the same impedances as in the last experiment, but 
arranged as in Fig. 173. As before stated, the voltmeter- 
ammeter-resistance method of solving inductive circuits is 
inapplicable to branched circuits ; so the wattmeter must 
be used as shown. 

Determine the equiva- 
lent impedance from 

Determine the angle 
of lag in the main circuit 
by 

COS ^ 







Determine the equiva- 
lent resistance, R (which 
is not the actual resistance of the parallel arrangement), 
from 

R = Z cos <^. 

Determine the equivalent reactance from 
X = Z sin </). 



TESTS. 



227 



All the constants of Coil i and Coil 2 are known ; but 
the resistance of the lamp had better be redetermined for 
the particular current used in it. 

Combine the admittances of these parts of the branched 
circuit into a polygon of admittances according to § 28. 
Take the reciprocal of the resulting admittance, — that 
is, the equivalent impedance, — and resolve it into its com- 
ponent parts of equivalent reactance and equivalent resist- 
ance. The actual plotting may be done on 25^' x 30'^ 
drawing-board to the scales 2 ohms = i cm. and i unit of 
admittance = 1000 cm. 

Make a report in the form of a table such as is used in 
the last experiment. The variation should not exceed 3%. 



109. Exp. 13. Calculation and Measurement of Result- 
ing Impedance of any Series-Parallel or Parallel-Series 
Arrangement of a Number of Impedances. — Arrange the 
apparatus as in Fig. 174, or according to any other scheme 
if it be desired to vary the experiment. 



^jmiM. ^sML. 

















«E_ 








9 




















1 




70 V. 








60'^ 


V100 

n 




's--, 

n 


A, 






i 1 


"- Q 












r 











Fig. 174. 



Determine the values of the resulting or equivalent 
jRf X, Z, and c^, as in Exp. 1 2. 



228 ALTERNATING-CURRENT MACHINES. 

Also determine the same quantities for the individual 
parts of the circuit under the conditions of use if they be 
not already known. 

In the graphic determination pursue the following 
steps : — 

1. Find the equivalent impedance of Coil 3, and the 

100 c.p. lamp, calling it M. 

2. Find the equivalent impedance of the 50 c.p. 

lamp, and M, calling it N. 

3. Find the equivalent impedance of Coil i, and 

Coil 2, calling it P. 

4. Find the equivalent impedance of P and N. This 

will be the required impedance of the whole 
circuit, and should be resolved into its com- 
ponent parts of equivalent resistance and 
equivalent reactance. Measure ^, the angle 
between the impedance and the resistance. 

Make a report in the form of a table as in the two pre- 
ceding experiments. The variation of the determinations 
by the two method should not exceed 3%. 

no. Exp. 14. Efficiency and Regulation of a Trans- 
former. — Arrange the apparatus as in Fig. 175. A 
two-throw switch allows the same voltmeter to read either 
primary or secondary pressure. The ammeter A.,^ may be 
used on the lower readings. The transformer used is 
the \ K. w. I to 2, stepping up from about 58 volts to 116, 
its rated range. It is operated at its rated frequency, 
60-. 

Increase the load from o to i k. w. (100% overload) by 
suitable steps. At each step take readings of the primary 
volts, primary watts ; secondary volts and secondary am- 



TESTS. 



229 



peres. Since the load is non-inductive, the product of the 
secondary volts and amperes gives the secondary watts. 

Determine the efficiency and the regulation, both in 
per cent, for each set of readings from 



f^Q efficiency 



% regulation = 



watts secondary 

-. ^ X 100. 

watts prnnary 

T volts prim. — volts sec. 
full-load sec. volts 



X 100. 



1 to 2 




Fig. 175. 

Plot t\vo curves on the same sheet, having as their 
common abscissae both watts and per cent of full-load 
secondary, and as their respective ordinates per cent effi- 
ciency and 100% — per cent regulation. 

III. Exp. 15. Determination of Load Losses in a 

Transformer The core losses are usually considered 

independent of the load, while all those that vary with the 
load are called the load losses. Their chief component is, 
of course, the PR loss in the copper, but there may be 
some eddy current and local hysteresis losses that vary 
with the load, and a determination of them all is made as 
follows : — 

Arrange the apparatus as in Fig. 176. The i to 2 



230 



ALTERNATING-CURRENT MACHINES. 



80 V.* 




Fig. 176. 



0.5 K. w. transformer is used with its low-tension side 
short-circuited. There will be but a small pressure gener- 
ated therein, and its current will demagnetize the core 

almost entirely ; hence 
all the losses measured 
may be considered as 
load, not core losses. 
Care must, of course, 
be taken to control the 
amount of current pass- 
ing through the trans- 
former. 

xAdjust the lamps so 
that about 1009^ overload current, 10 amperes, is shown 
by A^^. Read the ammeter and the wattmeter. Reduce 
the current by a suitable amount, and read again. So 
continue down to zero amperes, substituting A^ for A^. 
when the readings on the latter become unsatisfac- 
tory. 

Plot a curve ^^ith load in amperes as abscissae and load 
loss in watts as ordinates. 

Take care that the wire short-circuiting the low- 
pressure coil is of low resistance, and has good contacts. 
Note also that the pressure leads from the wattmeter 
should go direct to the terminals of the transformer, as, 
in general, the resistance of the wires leading to it is 
not negligible in comparison with the resistance of the 
coil itself. 

If the current exceeds the ampere capacity of the watt- 
meter, it is advisable to put in a single-pole switch to 
short-circuit the current coil at all times save when a 
reading is being taken. 



TESTS. 231 

112. Exp. 16. Determination of Core Losses of a 
Transformer, and Construction of an Efficiency Curve. — 

The core losses, hysteresis chiefly, are constant at all 
loads. Hence, the energy supplied to a transformer when 
its secondary is open-circuited, is practically a measure of 
these losses. 

Connect a wattmeter in the primary circuit of the 
i K. w. transformer when its primary is supplied with 
pressure at its rated voltage and frequency, and its sec- 
ondary is open-circuited. 

The wattmeter reading is the core loss. 

From a knowledge of the core loss and the load losses 
at various loads, construct an efficiency curve for various 
loads from o to 100% overload (secondary), the efficiency 
in per cent at any load P^ being 

F, - (Fr +F:) 



Px 



X 100, 



where P^ and P^ are the load and core losses respectively 
at the load P^. 

This curve should be similar to the efficiency curve 
found in Exp. 14. 

113. Ex. 17 and 18. Simultaneous Pressure and Cur- 
rent Curves from Primary and Secondary of a Trans- 
former. — It is desired to get these curves for two con- 
ditions. First, Exp. 17, with a full non-inductive load, 
and second, Exp. 18, with an equal (in amperes) very 
inductive load. 

Arrange apparatus as in Fig. 177. For the non- 
inductive load, lamps are suitable, for the inductive load 
the primaries of unloaded transformers are good ; and to 
get a nice adjustment Coil i can be put into circuit, and 



232 



ALTERNATING-CURRENT MACHINES. 



the current in it adjusted by moving its iron core in or 
out. 

It might be here remarked that if the transformer is 
supphed with current from a rotary converter, the E.M.F. 
balance described in Exp. 2 cannot use direct current from 
the same source as that which runs the converter, even 
though they be put on opposite sides of a three-wire 
system. A separate source of direct E.M.F. mAist in such 
case be suppHed for the balance, either from a separate 
direct-current generator or from a sufficient number of 



firi'i-iG 



To Load 




ind Balance 



Fig 177. 



cells of storage battery. If, however, the alternating cur- 
rent be passed through another transformer before being 
applied to the one under test, this trouble does not arise ; 
but the introduction of the second transformer has a dis- 
turbing effect on the wave-shape. 

It will be seen that the apparatus is merely an elabora- 
tion of that in Exp. 4, a four-way double-pole switch being 
used instead of the two-throw switch of the former experi- 
ment. This switch is conveniently made of mercury cups 
in a block of wood. For the i k. w. transformer at 58 to 
116 volts, the reading on A^^^ should be about 4.4 amperes. 



TESTS. 233 

Suitable values for the non-inductive resistances are, R^ = 
2,2 ohms, R^ — 4.4 ohms. These must be able to carry 
the currents without overheating, and must not be allowed 
to change their resistance due to change of temperature. 

Proceed as directed in P2xp. 4, taking readings every 
twelve electrical degrees throughout half a cycle. Take 
all four readings before moving the contact-maker. 

Plot the four curves of Exp. 1 7 on one paper, and those 
of Exp. 18 on another. In each case degrees will be 
the common abscissae. Both pressure curves of either 
experiment must be plotted to one scale of ordinates, both 
current curves to another. Careful work will show a 
phase difference slightly less than 180° between the 
primary and secondary pressures and currents, particularly 
in Exp. 18. 

114. Exp. 19. Calculation and Measurement of the 
Mutual-inductance of Transformer Coils at No Load. — 

(a) Measure the self-inductance of the primary and 
of the secondary coils by the method of Ex. 5, first part, 
the coil not under test being left open-circuited. 

(b) Do the same by the method of Exp. 5, second part. 
The results in the two cases should be alike if attention is 
paid to the following point. In measuring the primary 
inductance, apply the rated voltage so that it will send 
the charging current. In measuring the secondary, adjust 
the impressed voltage so that just such a current will flow 
as will give the same ampere-turns in the secondary as 
there were in the primary when it was being measured. 
If this precaution be not taken, the results will be changed 
by the effects of varying load, according to Exp. 7. 

(c) Using the average of the values of L^ and L^ as 



234 ALTERNATING CURRENT MACHINES. 

found in {a) and {b), calculate the value of J/ on the sup- 
position of no magnetic leakage, from 



id) Measure the mutual induction by the method of 
Exp. 8 a, taking care that the ampere turns are the same 
in each case, and the same as were used in {a) and {b). 

This last result may be slightly less than that arrived at 
in ic) because of magnetic leakage. 

Care should be taken throughout that the frequency be 
kept constant. 

115. Exp 20. Practice in Three-Phase Transformer 
Connections. — Three similar i to i transformers may be 
conveniently used for this experiment. The student may 
have to exercise some ingenuity in determining the direc- 
tion of winding in the coils. When each of the following 
connections has been made, excite the primaries by a three- 
phase current, and measure the pressure between each of 
the secondary wdres, seeing that all three sides have the 
same and the expected voltage, 

{a) Connect both primaries and secondaries in Y, as 
shown in Fig. '^'^. See that E^ = E^^. Then make the 
secondary a three-phase, four-wire system, and see that the 
voltage between any outside wire and the middle wire is 

{b) Connect both primaries and secondaries in A, Fig. Zjy 
and see that E^ — E^. Disconnect one transformer from 
the circuits, and observe that the three-phase pressure is 
still maintained in the secondary. 

if) Connect the primaries in Y, and the secondaries in 

E 
A, as in Fig. 90. Observe that E^ = — ^. 

V3 



TESTS. 235 

{d) Connect the primaries in A, and the secondaries in 
Y, as in Fig. 89, with a four-wire secondary system. Ob- 
serve that, with reference to the outside wires, E^ = V3 E^, 
while considering any outside wire and the middle wire, 

116. Exp. 21. External Characteristic of an Alter- 
nator Run the alternator at normal speed and field 

excitation, both being kept constant during the experiment. 
Arrange a variable non-inductive load — lamps preferably — 
so that readings can be taken from o load to 50% overload 
at suitable intervals. At each step note the armature 
current and the terminal pressure. 

Plot a curve with currents as abscissae, and pressures as 
ordinates. 

117. Exp. 22. Field Compounding Curve of an Alter- 
nator Run the alternator at constant rated speed. 

Arrange a variable non-inductive load of lamps, ranging 
by suitable steps from o load to 50% overload ; at each 
step adjust the field current, so that the rated terminal 
voltage is maintained. Take simultaneous readings of field- 
current and armature current. 

Plot a curve with armature currents as ordinates, and 
field currents as abscissae. 

Note that the speed must be kept constant, that the 
terminal pressure must be kept constant, and that read- 
ings should be taken only with ascending values of field 
currents, as magnetic retentivity will distort the curve 
somewhat if the field current is run too high, and then 
brought down to the required point. 

118. Exp. 23. No-Load Saturation Curve of an Alter- 
nator. — Run the alternator at constant rated speed, and 



236 



ALTERNATING-CURRENT MACHINES. 



excite the fields from zero up to full excitation, taking, at 
suitable intervals, readings of the field current, and the 
no-load armature voltage. Repeat, carrying the excitation 
from full excitation down to zero. 

Plot the two curves on one sheet, using field currents as 
abscissae, and terminal pressures as ordinates. The two 
curves will not exactly coincide, because of the magnetic 
retentivity of the iron. 

Care must be taken always to adjust the field current by 
increasing from a lower value to a higher when taking the 
ascending curve ; and by decreasing from a higher value to 
a lower when taking the descending curve. 

119. Exp. 24. Full-Load Saturation Curve of an Al- 
ternator. — Arrange apparatus as in Fig. 178. The alter- 
nator is a IK. w. 80 volt, single-phase machine. The 
machine is given a non-inductive load of lamps, and a 
heavy current rheostat which has zero resistance on the 




Fig. 178. 



last point. Run the alternator at its rated speed. Make 
the resistance of the external circuit zero — i.e., short-circuit 
it through the ammeter. Adjust the field rheostat to its 
maximum resistance, and close the field switch. Increase 
the excitation by manipulating the field rheostat until the 
rated full-load current is flowing in the external circuit, as 



TESTS. 



237 



shown on A^^. Take readings of the field amperes, and the 
terminal volts, the latter being zero at this step. Increase 
the resistance of the armature circuit by a suitable amount, 
and readjust the excitation till the rated full-load current is 
again flowing in the armature, and take readings of field 
current and terminal voltage. Repeat at suitable steps 
until full field excitation is obtained. 

Plot a curve on the same paper, and to the same scale as 
that of Exp. 23, using field currents as abscissae, and 
terminal volts as ordinates. 

Take heed that readings are always taken with ascend- 
ing values of field current, and when the ammeter in the 
armature circuit shows rated full-load current. The speed 
must be kept constant. 



120. Exp. 25. Synchronous Impedance of an Alterna- 
tor. — As stated in § 38, the synchronous impedance of an 
alternator varies somewhat with the load, but is practically 
constant at all excitations ; hence its determination is easily 
accomplished in the following manner : — 

Arrange the apparatus as shown in Fig. 179. Run the 
alternator at its rated speed. By means of the field rheo- 
stat cut the excita- 
tion down to a mini- f<p==^\A, 
mum. Short-circuit 
the armature 
through an am- 
meter and switch 
as shown. Adjust 
excitation so that 
the ammeter shows about -J- full-load current, and note the 
ammeter reading. Open the switch in the armature cir- 




Fig. 179. 



238 ALTERNATING-CURRENT MACHINES. 

cuit and note the terminal volts. Close the switch, read- 
just excitation so that the load is increased by a suitable 
amount, and repeat the readings. Repeat until a limit is 
reached, either because full field excitation has been ob- 
tained, or because the machine is being too severely over- 
loaded. Which of these two conditions arises first depends 
upon the synchronous impedance of the machine. 

Calculate the synchronous impedance for each set of 

readings from 

open circuit voltage 



Syn. Imp. 



short circuit current 



when the readings are for the same excitation and speed. 

Plot a curve with armature currents as abscissae, and the 
values of the synchronous impedance as ordinates. 

Particular attention should be paid that the speed be 
kept constant, as it is liable to rise on throwing off the 
load. 

121. Exp. 26. Core loss of an Alternator The core 

loss of any armature is determined by measuring the dif- 
ference in power required to run it with and without field 
excitation. With an alternator this is most easily done by 
running the armature by a rated motor and observing the 
power input thereto. It is desirable to have the quantity 
sought as large as possible in comparison with the quanti- 
ties observed ; hence the rated motor used should be as 
small as is practicable. 

The alternator must be driven at its rated speed, and the 
pulleys so proportioned that the motor will run at its rated 
speed also ; or else a special efficiency curve of the motor 
must be obtained for the speed at which it will be required 
to run. 



TESTS. 239 

A wattmeter placed in the motor circuit will indicate 
the power input thereto ; or, if it be a direct-current motor, 
a voltmeter and ammeter can be used. 

Let ^ = watts input to motor when the alternator field 

is not excited, 
and ;;z = efficiency of motor at this input. 
Let ^ = watts input to motor when the alternator fields 

are fully excited, 
and ;/ = efficiency of motor at this input. 

Then the core loss in watts is 

P^ = Bii — Am. 

It is well to repeat the measurements a number of times 
and average the results. 

Since the losses in shafting and belting are practically 
the same at all loads, these do not affect the accuracy of 
the results. 

122. Exp. 27. Complete Test of a i-H.P. Three-Phaso 

Induction Motor As a test of the motor performance 

solely, the voltage at the motor terminals should be kept 
constant throughout the test. This may easily be accom- 
plished if the motor is run from a separate alternator. If, 
however, it is run from an inverted converter, and particu- 
larly if the desired voltage has to be obtained by transfor- 
mation, there will be a slight drop of voltage as the load 
increases. 

Since the power-factor in this test will run from very 
low to about 80%, the method of measuring three-phase 
power shown in Fig. 169 will be used, as it requires but 
one instrument reading, and leads to no uncertainty as to 



240 



ALTERXATIXG-CURREXT MACHINES. 



algebraic signs. Tlie apparatus used is simply an amme- 
ter, a voltmeter, and a wattmeter connected into the motor 
circuit. Fig. 180 shows the arrangement, the two watt- 
meters not being used at once, but being alternative, — one 
for high readings, the other for low, — thus securing a 
greater accuracy over a wide range. The motor when 
stalled takes about 1 8 amperes ; so this is its momentary 




Fig. 180. 

starting current. Care must be taken in starting up that 
the measuring instruments are not injured by such a flow 
of current. The larger wattmeter has a capacity of 
2.5 K. w., and a 2 5 -ampere limit; the smaller a capacity 
for 300 watts, and a 5 -ampere limit. Either of these, as 
well as the voltmeter, will stand the rated motor pressure, 
— 1 10 volts. 

The power output of the motor is absorbed in a strap 



TESTS. 



241 



brake, as shown in Fig. 181. With a 4.5'^ pulley at 1800 
revolutions the spring balances should have ranges of 
about 30 lbs. and 4 lbs. respectively. 

The motor must be supplied with current at its rated 
voltage and frequency, and the frequency must be kept 
constant throughout the experiment. 

Observations. — Take readings at suitable 
intervals, — say steps of 4 lbs. each on the 
larger scale, — from no load to the stalling 
of the motor. Do not leave the motor 
stalled, as it overloads the instruments. 

At each step take readings of the watt- 
meter, ammeter, voltmeter, both spring 
balances, and the speed of the motor. 

Repeat the experiment three times with 
fifteen-minute intervals between the repeti- 
tions. The scale readings, P, will be the 
same, at any one step, for all three trials, and the other 
values can be averaged to partially ehminate errors of 
observation. 

Calculations. — Using the average values of the three 




Fig, 181, 



readings at any one step, fill 



out the following table 



I 


2 


3 


4 


s 


6 


7 


8 


9 


H 

D 

CL. 

H 

H 
H 
< 


D 

Pm 

Z 

H 

< 


< 

< a 


> 


(I) 

< 

X 
Oh 

s 

a. 
S 
< 


W 
a. 
S 
< 

J 

> 




u 

< 

tlH 

1 


u 
u 

lb 


>< 
u 
z 

a. 


a. 





















242 ALTERXATIXG-CURRENT MACHINES. 

— VtriP-P') 

12 

(i) Watts output = 746, 

33,000 

where 

d = diameter of pulley in inches. 

F = revolutions per minute. 

{F — P') = difference in scale readings in pounds. 

(2) Watts input = 3 X wattmeter reading. 

(3) Volts at terminals = voltmeter reading. 

(4) Amperes per phase = ammeter reading. 

(5) Volt-amperes input = V3 X volts at terminals X amperes 

per phase. 

/^N -r. r y Watts input 

(6) Power-factor % = =^-, ^ X 100. 

^ ' \ olt-amperes input 

/ N -i-rc • ^ Watts output 

(7) Efficiency % = — ^ — -. — ^-— x 100. 



atts input 

/a; 

-amperes input 

(9) Slip % = 

where 



/ox . ^ rr • Watts output 

(8) Apparent eiiiciency = — -; -. X 100. 

^ ^ "^ \ olt-amperes input 

60/ ' 



V == revolutions per minute, 

/ = frequency, 

/ = number of pairs of poles. 

Plottijig of Curves, — Plot eight curves on one paper. 
All the curves will have watts output as abscissas. The 
points of 25%, 509^, TS'^lc 100%, and 125% of full load 
should also be indicated. 

The ordinates for the first seven curves are taken from 
columns 2 to 8 respectively. 



TESTS. 



243 



The ordinates for the last curve are found by subtract- 
ing the per cent slip from 100%. 

Curves should be marked with the names appearing at 
the heads of the columns from which their ordinates were 
taken. The curves from columns 2 and 5 should be to 
the same scale of ordinates. Those from columns 6, 7, 8, 
and 9 will all have the same scale of ordinates, which will 
be per cents, and should run from o to 100%. 

There will thus be four scales of ordinates, and they 
should be marked respectively, ''Watts or Volt Amperes," 
"Volts," "Amperes," and "Per Cent." On the margin 
state the name and size of the machine, and the date of 
test. 



123. Exp. 28. Complete Test of a i H.P., Three-Phase 
Induction Motor, run on a Single-phase Circuit Through a 
Condenser-Compensator. — The function of the condenser- 
compensator was discussed in § 6"j. The arrangement of 
apparatus is shown in Fig. 182. Another wattmeter and 



J 



115 V 
60 f\J 




?&1 



Tt 



V200 



9smmm 



To 



Motor 



Fig, 182. 

another ammeter may be used to alternate with those 
shown to secure greater accuracy in the lower ranges if 
considered advisable. 

The same absorption dynamometer is used as in Exp. 
27 ; and the directions there given for taking observations, 
and for calculating and plotting results, should be followed 
with the following exceptions : take readings at 2-lb. steps, 



244 ALTERNATING-CURRENT MACHINES. 

since the motor is half the size of the other ; since this is 
single-phase, the wattmeter readings go direct in column 
2, and the products of the volts by the amperes go direct 
in column 5. 

It may be found that the capacity of the condenser- 
compensator has been so proportioned that in plotting 
the results, curves 2 and 5, and also 7 and 8, will be nearly 
coincident, and that curve 6 is practically a straight hori- 
zontal line. 

124. Exp. 29. Methods of Synchronizing. — Synchro- 
nous motors and also converters must be synchronized 
before being connected to the mains from which they 
receive their power. There are a number of ways of 
doing this, of which the best depends upon attendant cir- 
cumstances, {a) The motor and generator may be elec- 
trically connected while at rest, and the latter started up 
slowly, the motor — not loaded — then starting up and 
running synchronously, {b) The field circuit of the motor 
may be left open, and the armature started up — without 
load — as an induction motor until near synchronism, and 
the field switch then closed. In large machines this 
endangers the insulation of the field coils, {c) The arma- 
ture may be brought to speed mechanically, either by a 
small direct connected induction motor or by a belt from 
some moving pulley, {d) In converters the machine can 
be started and brought to speed from the direct-current 
end like a direct-current motor, if there be direct cur- 
rent available. This requires a starting-box and a field 
rheostat. 

The two convenient methods for synchronizing the 
I K.w. three-phase converter are {b) and {d) \ the former 



TESTS. 



245 



will be practiced in Exp. 30, the latter is the subject for 
the present experiment. 

Arrange the apparatus as shown in Fig. 183. At 
starting, the field coils of the converter must be excited 
from the source of direct current, but when running as 
a converter the machine must be excited from its own 
brushes. This necessitates the two switches, a and b. 
These switches must not be both open at once, at least 
while the machine is running from the direct-current end ; 
and if they are not rightly connected the direct-current 



Main Switch 



3 Phase 




Fig. 183. 

source will be short-circuited when they are both closed at 
once. It is best, after the set-up is made, to test across 
the switches with a voltmeter. The switch must not be 
closed if any pressure shows across the gap it is intended 
closing. 

When the connections have been properly made, open 
the main switch and switch a, close switch b and the 
switch from the direct-current source of supply, and start 
the machine up as a direct-current motor. When the 
starting-box is completely on, first close a, then open b. 



246 ALTERNATING-CURRENT MACHINES. 

Then manipulate the field rheostat until the machine 
reaches synchronism. The synchronizing lamps will all 
be dark at once when the machine is in step. When 
the periods of darkness become quite long, say several 
seconds, the main switch can be closed, the switch from 
the direct-current source be opened, and the machine will 
be running as a self-excited converter. 

If all the lamps do not get dark at once, but two stay 
lighted while the other is dark, the generator currents are 
in such directions as to tend to reverse the direction of 
rotation of the converter armature. Two of the leads in 
the alternating-current circuit should then be transposed. 
It might here be noticed that if an inverted converter be 
used as a source of alternating currents, and it be desired 
to synchronize another converter with it as described 
above, and if an Edison three- wire system be the common 
source of direct-current supply to these machines, then 
care must be taken that both converters are connected to 
the same side of the system. If they be connected 
to opposite sides, then when the machines are in step 
there is a pressure of 117 volts across the main switch, 
and closing the latter would naturally cause the blowing 
of some fuse. 

125. Exp. 30. Variation of Lag or Lead of Current 
in a Three-Phase i K.W. Synchronous Motor. — Arrange 
the apparatus as in Fig. 184. Either the 3 or the 15 
ampere ammeter can be put in circuit, according to the load. 
Remember that in starting up there will be an excessive 
flow of current. 

The direct-current brushes on the machine may be 
removed for this experiment. Synchronize the motor by 



TESTS. 



247 



letting it run as an induction motor till near synchronism. 
Then close the field switch, having first adjusted the 
rheostat, so that about normal field current will flow. The 
machine will fail to go into step if this adjustment is not 
made. Premature closing of the field switch is also a 
cause of failure to synchronize. It is easy to tell if the 
machine goes into step or not. If it does, the current in 
the armature circuit goes down ; if it does not, the machine 
slows down and even stops, and the current goes up, 

(a) When the motor is synchronized, reduce the field 
current as far as possible, without losing step. In some 



3 Phase • 



•^500 

vwvvv\ 




Pig. 184. 

machines it may be reduced to zero, the residual magnet- 
ism affording enough field to keep the armature syn- 
chronized. From this point increase the field current by 
suitable steps to the maximum allowable current, or until 
the machine loses synchronism. At each step note the 
field current and the armature current. Then from the 
maximum field current, decrease to the minimum by 
the same steps, taking readings again of the two ammeters. 
Due to magnetic retentivity the two curves will not 
coincide. 

Plot the two curves on one paper, using field currents 
as abscissae, and armature currents as ordinates. 



248 ALTERNATING-CURRENT MACHINES. 

That excitation at which the no-load armature current is 
a minimum, is called the normal excitation of a syn- 
chronous machine. 

{b) Repeat the foregoing, save that a strap-brake load is 
applied to the motor. This load should be adjusted to 
about 75% of full load, and left constant. It will be 
found that the motor will not submit to so wide a 
range of field currents when loaded as when running 
light. 

Plot these curves on the same sheet. 

126. Exp. 31. Commercial Efficiency of a Synchron- 
ous Motor. — The same arrangement of the same appa- 
ratus is here used as in Exp. 27, save that it is applied to 
a three-phase synchronous motor, whose field coils must 
be separately excited from a direct-current source, and 
with a suitable rheostat in series. 

Synchronize the motor by the method of Exp. 30, being 
careful that the excessive starting current does not pass 
through the coil of the low-reading wattmeter. When the 
armature is in step, adjust the field rheostat to give normal 
excitation ; i.e., so that the armature current is a minimum. 
This adjustment must not be changed during the experi- 
ment. The frequency should be kept constant, and the 
voltage also if possible ; if not, account should be taken of 
its fall, and a voltage curve drawn on the same sheet as the 
efficiency curve. 

Load the m.otor by the strap-brake shown in Exp. 2y, 
increasing by suitable steps from zero till the motor falls 
out of step. At each step read the wattmeter and the 
two spring balances. Repeat the experiment three times, 
each time stopping at the same points on the larger spring 



TESTS. 249 

balance, so that the other values can be averaged, reducing 
errors of observation. 

Plot a curve with watts output and per cent of load as 
abscissae, and per cent efficiency as ordinates. 

127. Exp. 32. Curves of Current and Power-Factor 
of a Synchronous Motor, with (a) Super-Excitation, and 

(d) Sub-Excitation The arrangement of apparatus is that 

of Exp. 27, applied to the synchronous motor. 

For the first part of the experiment, the fields should be 
excited by a current about 50% greater than the normal 
field current, and for the second part by a current about 
50% less. The frequency and the voltage should be kept 
constant. 

For each part load the motor with the strap-brake, in- 
creasing from zero by suitable steps till the armature falls 
out of synchronism. At each step take readings of the 
ammeter, voltmeter, and wattmeter, as well as of the two 
spring balances. Repeat each part three times, averaging 
the results. 

Tabulate the results under columns headed " Watts Out- 
put," "Watts Input," "Volt-amperes Input," and "Power- 
factor." 

Plot on one paper the curves for the super-excited 
condition, making watts output the common abscissae, and 
armature currents and power-factors respectively the 
ordinates. 

Plot on another sheet similar curves for the sub-excited 
condition. 

128. Exp. 33. External Characteristic of a Converter, 
A.C. to D.C. with Self -Excitation. — Arrange apparatus 
as in Fig. 183, Exp. 29. When the converter is running 



250 ALTERNATING-CURRENT MACHINES. 

from the alternating end, and free from the source of 
direct-current supply, adjust the field rheostat to that 
point that gives a mmimum armature current at no load. 
If this point is not already known, it will be necessary 
to put the 15-ampere ammeter in one of the alternating- 
current mains to determine it. 

When the above conditions are fulfilled, load the direct- 
current end of the converter with lamps, from o up to 50% 
overload (say 15 amperes), by steps of about one ampere 
each. At each step take readings of the armature current 
and the terminal pressure, using the standard direct-current 
instruments for the purpose. - 

Plot a curve with armature currents as abscissae, and 
terminal pressure as ordinates. 

129. Exp. 34. Efficiency of a Converter from A.C. to 
D.C. — The arrangement of apparatus is that of the last 
experiment with the addition of a wattmeter suitably con- 
nected in the alternating current mains to measure the 
power input. In fact, all the necessary data for Exp. 33 
are incidentally secured in the course of this experiment. 

Run and excite converter as in the last experiment. 
The frequency and the voltage should be kept constant. 
The direct-current end is to be loaded with lamps by suit- 
able steps from o to 50% overload. The w^atts input can 
be determined from the wattmeter reading, the watts out- 
put from the product of the voltmeter and the ammeter 
reading. 

Plot an efficiency curve. 

Note. — The brush friction of the direct-current brushes may be so 
great that the converter cannot be synchronized by the metliod of starting 
as an induction motor, the slip being so great as to prevent its picking up 



TESTS. 251 

when the field circuit is closed. In such case either the direct-current 
brushes must be temporarily removed, or the machine must be synchronized 
by some of the other methods given in Exp. 29. If the converter hunts so 
badly as to interfere with the instrument readings, it may be because the 
direct-current brushes are not in proper adjustment, and the selection of a 
better commutating plane will remedy the trouble. 

130. Exp. 35. Efficiency of an Inverted Converter. — 

Since it is inconvenient to put a variable, non-inductive, 
balanced load on a three-phase circuit, the single-phase 
rings wil] be used in this experiment. 

The arrangement of apparatus requires a direct-current 
ammeter and voltmeter on the D.C. end, and a wattmeter 
on the A.C. end. Or two wattmeters can be used if avail- 
able. The converter is started from the D.C, end by 
means of a starting-box, the field rheostat is adjusted until 
the armature is running at its rated speed. This speed 
must be kept constant during the test by manipulating the 
rheostat. A non-inductive load is applied to the A.C. end, 
increasing by suitable steps from o to 50% overload. 

Plot an efficiency curve and an external characteristic 
curve. The latter will approximate a straight horizontal 
line, being much better than that secured in Exp. 33, be- 
cause, in this case, the field current is unaffected by any 
drop of voltage in the armature. 



INDEX. 



[The figures refer to page numbers.] 



Admittance of circuit, 45. 
Ageing of iron, 123. 
Air-blast transformers, 128. 
All-day efficiency, 139. 
Alternating current, definition of, i. 
Alternations, definition of, i. 
Alternators, Bullock, 89. 

core loss of, 238. 

efficiency of, 72. 

external characteristic of, 235. 

field compounding of, 235. 

General Electric Co.'s, 76,78,87. 

inductor, 80. 

load losses of, 73. 

parallel running of, 168. 

revolving field, 85. 

saturation curve of, full-load, 
236. 
no-load, 235. 

single-phase, 57 

Stanley, 81. 

Warren, 85. 

Westinghouse, 76, 91. 
Aluminum line wire, 1S9. 
Angle of lag or lead, 10. 
Average values of pressure and cur- 
rent, 8. 
Apparent resistance, 25, 45. 
Armature, E.M.F. generated in, 62. 

inductance, 68. 

reaction, 67. 



Armature : 

windings, 65. 
Auto-transfoiTner, 94. 

connections of, 121. 

Brake, Prony, 241. 
Bullock alternators, 89. 

Calculation of leakage flux, 108. 

resulting impedance, 50, 224. 
Capacity, measurement of, 213. 

of circuit, 45. 

of condenser, 29. 
formula for, 31. 

of transmission line, 193. 

reactance, 40, 45. 

unit of, 30. 
Centrifugal clutch pulley, 152. 
Characteristic of alternator, 235. 

of converter, 249. 

of inverted converter, 251. 
Chemical solution to detect current. 

204. 
Choke coils, 28. 
Circuits, time constant of, 21. 

with R, Z, and C, 40. 
Coefficient of leakage, 71. 

of saturation, 70. 

of self-induction, 16. 
Combined method, measuring power, 
218. 



= 53 



254 



INDEX. 



Commercial efficiency of synchro- 
nous motor, 248. 
Compensated winding, 78. 
Compensators, 94. 

connections of, 121. 
Composite winding, 74, 76. 
Compounding curve of alternator, 

235- 
Condenser, hydraulic analogy, 37. 
compensator, 154. 
capacity of, 29. 

formula for, 31. 
electrolytic, 31. 
Condensers, 29. 
in parallel, 32. 
in series, 33. 
Conductance of circuit, 46. 
Connections of transformers, 115. 
Constant current transformers, 130, 

potential, regulation for, 57. 
Converter, 169. 

armature heating, 175. 

reaction, 177. 
current relations in, 173. 
efficiency of, 250. 
E.M.F. relations in, 171, 
external characteristic of, 249. 
inverted, 170. 

efficiency of, 251. 
external characteristic of, 251. 
regulation of, 179. 
starting of, 177. 
Cooling of transformers, 128. 
Copper loss in transfoimers, 104. 

weight of, for lines, 197. 
Core flux in transforaner, 92, 125. 
loss of alternator, 74. 

measurement of, 238. 
of transformer, 99. 
measurement of, 231. 
type transformer, 92, 125. 



Zow^\&\ E.M.F. of self-induction, 26. 
Current and pressure relations : 
in a condenser, 39. 
transformer, determination of, 
231. 
average value of, 8. 
effective value of, 7. 
flow, foiTnula for, 42. 
instantaneous value of, 4. 
lag or lead of, 10. 
magnetic energy of, 23. 
produced by harmonic E.M.F. , 
24. 
Curve, saturation, full load, 236. 
no load, 235. 
sine, 4. 

form-factor of, 9. 
Curves from transformer, 231. 

of current and power-factor in 
synchronous motor, 249. 
E.M.F, actual, 6. 
distortion of, 9. 
determination of, 205, 208. 
field compounding of alterna- 
tor, 235. 
Cycle, definition of, i. 

Decay of current in circuit, 21. 
Definitions of terms, 44. 
Delta or mesh connection, 61. 

of transformers, 119. 
Design of transfoimer, 133. 
Dielectric hysteresis, 31. 
Dielectrics for condensers, 30. 
Distribution constant, 64. 

Eddy current loss, 99. 
Effective values of current and pres- 
sure, 7, 
Efficiency, all day, 105. 
of alternator, 72. 



INDEX. 



255 



Efficiency : 

of converter, 250. 
of inverted converter, 251. 
of synchronous motor, 248. 
of transformer, 104, ri5, 139. 
measurement of, 228. 
E.M.F., counter, of self-induction, 
26. 
generated in armature, 62. 
E.Af.F., wave, shape of, 3, 6. 
determination of, 205. 
E.M.F.''s in series, 46. 
Energy of a started current, 23. 
Equivalent F, X, and Z of trans- 
former, 97. 
leakage inductance, 108. 
Exact solution of transformer, iii. 
Exciting current of transformer, 94, 

103. 
External characteristic of alternator, 

235- 
converter, 249. 
inverted converter, 251. 

Farad, definition of, 30. 
Field compounding curve of alter- 
nator, 235. 
. rotating, 141. 
Plux density in transformers, 95. 
Form-factor, 9. 

determination of, 206. 
Formula for current in any circuit, 

42. 
Four-phase currents, 11, 

systems, 59. 
Frequency changers, 157. 
definition of, i. 
determination of, 2. 
Frequencies, for power transmis- 
sion, 183. 
standard, 2. 



Full-load saturation curve, 70. 
determination of, 236. 

General Electric Co.'s alternator, 
76, 78, 87. 
regulator, 181. 
induction motor, 146. 
transformer, 125, 132. 
Growth of current in condensive 
circuit, 34. 
in inductive circuit, 20. 

Harmonic E.M.F., current produced 
by, 24. 

shadowgraph, 3, 
Henry, definition of, 16. 
Hydraulic analogy of condenser, 37. 
Hysteresis, dielectric, 31. 

loss in alternators, 73. 

transformers, 100. 

Impedance, definition of, 25. 

of circuit, 45, 224-228. 

synchronous, 69. 

measurement of, 237. 
Impedances in parallel, 226. 

in series, 224. 
Inductance, measurement of, 211. 

mutual, measurement of, 216. 

of circuit, 45. 

of transmission lines, 191. 

self, described, 15. 

unit of self, 16. 

variation with load of, 214. 
Inductances, practical values of, 18. 
Induction motors, 142. 

behavior of, 149. 

General Electric Co.'s, 146. 

single phase, 154. 

slip of, 144. 

speed regulation, 157. 



256 



INDEX. 



Induction motors : 

starting of, 150. 

tests of, 239, 243. 

^Yagner, 155. 

\\^estinghouse, 143. 
Inductive reactance, 25, 45. 
Inductor alternators, 80. 
Instantaneous values of current and 

pressure, 4. 
Inverted converter, 170. 

Lag of current, 10. 
Lead of current, 10. 
Leakage coefficient, 71. 

flux, 106. 

inductance, 108. 
Lighting transformers, 122. 
Line capacity, 193. 

constants (Table), 194. 

inductance, 191. 

loss, curves of, 196. 

resistance, 190. 

wire, aluminum, 189. 
Linkages defined, 16. 
Load losses in alternator, 73. 
in transformer, 99, 
measurement of, 229. 
Logarithmic change of current, 20. 
Losses in synchronous machines, 73. 

in transformers. 99. 

in transmission Hues, 1S6. 

Magnetic energy of current, 23. 
Magnetizing current of transformer, 

102. 
Magnitude of self -inductance, 19. 
Measurement of capacity, 213. 

of core loss in transformer, 231. 
of core loss in alternator, 238. 
of efficiency of transformer, 
228. 



Measurement : 

of load losses in transformer, 

229. 
of mutual induction, 216. 
of transformer coils, 233. 
of power, pol}-phase, 219. 
of power, single phase, 217. 
of regulation of transformer, 

228. 
of resulting impedance, 224, 

226, 228. 
of self-inductance, 211, 
Mershon balance, 206. 
Mesh or delta connection, 61. 

of transformers, 119. 
Methods of connecting transform- 
ers, 115. 
of synchronizing, 244. 
Microfarad, definition of, 30. 
Monocyclic system, 156. 
Motor, induction, 142. 
behavior of, 149. 
General Electric Co.'s, 146. 
measurement of efficiency, 

239' 243. 
single-phase, 154. 
speed regulation, 157. 
starting of, 150. 
induction, treatment by trans- 
former method, 147. 
Wagner, 155. 
Westinghouse, 143. 
starters, General Electric Co.'s, 

151- 
Westinghouse, 151. 
synchronous, 158. 

measurement of efficiency, 
24S. 
Mutual induction, measurement of, 
216. 
in transformer coils, 233. 



INDEX. 



257 



Natural draft transfonners, 128. 
No-load saturation curve, 70. 
determination of, 235. 
Number of phases for transmission, 
189. 

Oil-cooled transformers, 129. 
Operation of induction motors, 143. 
Operati\-e range of synchronous 
motors, 161. 

Parallel circuits, 226. 
Parallel running of alternators, 168. 
Parallel-series circuits, 227. 
Parallelogram of E.M.F.'s, 27, 48. 
Peculiarities of A.C. circuits, 203, 
Phase, 10. 

relations in condensive circuits, 

splitters, 153. 
Phases, number of, for transmission, 

189. 
Polygon of admittances, 53. 
of E.M.F's, 49. 
of impedances, 50. 
Polyphase alternators, 58. 
currents, 12. 

power, measurement of, 219. 
Power curves, determination of, 
209. 
factor, definition of, 14. 
in A.C. circuits, 12. 

measurement of, 217-224. 
transmission, 182. 
Power transmission, frequency for, 
183. 
voltage for, 185. 
phases for, 1 89. 
Practical values of inductances, 1 8. 
Pressure and current relations in 
condenser, 39. 



Pressure and current relations in 
transformer, determination of, 
231. 
average value of, 8. 
curves, actual, 6. 

causes of distortion, 5. 
determination of, 205, 208. 
effective value of, 7. 
for transmission, 185. 
instantaneous value of, 4. 
Primary of transformer, 92. 
Prony brake, 241. 

Quarter-phase currents, 4. 
systems, 59. 

Ratio of transformation, 92. 

in induction motor, 148. 
Reactance of any circuit, 45. 
of condensive circuit, 40. 
of inductive circuit, 25. 
synchronous, 69. 

measurement of, 237. 
Regulation for constant potential, 

74- 
of converters, 179. 
of transformers, 106, 139. 
mieasurement of, 228. 
Regulator, General Electric Co.'s, 
181. 
Stillwell, 180. 
Relations of current and pressure in 
condenser, 39. 
in transformer, determination 
of, 231. 
of E.M.F.^s in converters, 173 
Resistance, apparent, 25, 45. 

of inductive circuits, 25, 44. 
of line wire, 190. 
Resonance, 42. 
Revolving field alternators, 85. 



258 



INDEX. 



Rotary converter, see Converter. 
Rotating magnetic field, 141. 
Rotor, definition of, 142. 

Saturation, coefiicient, 70. 
curves, 70. 

of alternator, full-load, 236, 
no-load, 235. 
Scott transformer, 118. 
Secondary of transformer, 92. 
Self-inductance, counter E.M.F. of, 
26. 

described, 15. 
measurement of, 211. 
unit of, 16. 
Series circuits, 224. 
Series-parallel circuits, 227, 
Shadowgraph, harmonic, 3. 
Shape of current wave, determina- 
tion of, 208. 
of E.M.F. wave, 3. 

determination of, 205, 209. 
Shell type transformer, 92, 123. 
Simultaneous curves from trans- 
former, 231. 
Sine curve, 4. 

form factor of, 9. 
Single-phase alternators, 57. 
current, 1 1 . 
induction motor, 154. 

test, 243. 
power, measurement of, 217. 
Sinusoid, 4. 

Slip of induction motors, 144. 
Skin effect, 191. 

Solution for detecting currents, 204. 
Squirrel-cage motors, starting of, 
150. 
rotor defined, 143. 
Standard frequencies, 2. 
Stanley alternator, 81. 



Stanley transformer, 127. 
Star or Y connection, 60. 

of transformers, 119. 
Started current, magnetic energy of, 

23- 
Starting of induction motors, 150, 
of synchronous motors, 155. 
Stator defined, 142. 
Step-up and step-down transform- 
ers, 93. 
Stilhvell regulator, 180 
Strap brake, 241. 
Susceptance of circuit, 46. 
Synchronizer, 167. 
Synchronizing, methods of, 244. 
Synchronous motors, 158. 
hunting of, 163. 
measurement of efiiciency, 

248. 
operative range, 161. 
starting of, 165. 
variation of lag or lead, 246. 
of current and power-factor, 
249. 
reactance, 69. 

measurement of, 237. 

Table of line constants, 194. 
Temperature, effect on core loss, 103. 
Test of induction motor, 239. 

with condenser compensator, 

243- 
Three-ammeter method for measur- 
ing power, 218. 
voltmeter method, 217. 
Three-phase induction motor test, 

239- 
power measurements, 221. 
systems, 60. 

transformations, 119,234. 
Time constant of circuit, 21. 



INDEX. 



259 



Transformation, ratio of, 92. 
Transformer, air blast, 129. 
connections of, 115, 234. 
constant current, 130. 
cooling of, 128. 
definition, 92. 
design of, 133. 
efiiciency, 104, 115, 139, 

measurement of, 228. 
exact solution of, iii. 
flux in, 94. 
for lighting, 122. 
General Electric Co.'s, 125, 132. 
losses, 99. 
measurement of core losses, 231. 

load losses, 229, 

mutual induction in, 234. 
method of treatment for induc- 
tion motors, 147. 
natural draft, 128. 
oil cooled, 129. 
regulation of, 106, 139. 

measurement of, 228. 
Scott, T 1 8. 

simultaneous curves from, 311. 
Stanley, 127. 
Wagner, 123. 
water-cooled, 129. 
Westinghouse, 126. 
Transmission of power, 182. 
lines, capacity of, 193. 

inductance of, 191. 

resistance of, 190. 

table of constants for, 194. 
losses in, 186. 

curves of, 196. 
Triangle of E.M.F.''s, 25, 40. 



Two-phase currents, 11. 

power measurements, 219. 

systems, 59. 
Type A. O. transformers, 127. 

H transformers, 125. 

M transformers, 123. 

O.D. transformers, 126. 

Variation in synchronous motor of 
lag or lead, 246. 

in current and power factor, 
249. 

of inductance with load, 214. 
Vibrating filament, 205. 
Voltage, average value, 8. 

curves of actual, 6. 

effective value of, 7. 

generated in armature, 62. 

Wagner induction motor, 155. 

transformer, 123. 
Warren alternator, 85. 
Water-cooled transformers, 129. 
Wattmeters in polyphase circuits, 

219. 
Wave-shape, 3. 

causes of distortion of, 5. 

determination of, 205, 208. 

form factor of, 9. 
Weight of copper for Hues, 197, 
Westinghouse alternator, 76, 91. 

induction motor, 143. 

transformer, 126. 

Y-connection, 60. 

of compensators, 121. 
of transformers, 119. 



LIST OF WORKS 



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ABBOTT, A. V. The Electrical Transmission of Energy. A Manual for the 
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ARSfOIiD, JE. Armature Windings of Direct-current Dynamos. Extension and 
Application of a General Winding Rule. Translated from the original German 
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ATKINSOX, PROF. A. A. Electrical and Magnetic Calculations. . . . $1.50 

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Elements of Electric Lighting, including Electric Generation, Measurement, 

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Power Transmitted by Electricity and applied by the Electric Motor, includ- 
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BAOT, F. B. Dynamo Tender's Hand-book. 70 Illustrations. 16mo, cloth, $1.00 
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CROCKER, F. B., and S. S. IVHEEEER. The Practical Management of 

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ElVIXO, J. A. Magnetic Induction in Iron and Other Metals. Second issue. 
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*** A Oeneral Catalogue— 80 pag-es— of TTorlis in all brancbes of 
Electrical Science furnisbed g^ratis on application. 



^J1^^ S 



\m^ 



MAY 8 1902 

1 COPY DEI. '^^-- 
MAY 8 1902 



